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Question 263251: I need help to solve the equation 2x - 3y = -7 using slope and y-intercept

Answer by ikleyn(53748)

(Show Source):

You can put this solution on YOUR website!
.

How the problem is worded in the post, this formulation is mathematically incorrect.

A correct formulation is THIS

    Transform this given equation to the slope-intercept form.

Question 1004808: Solve:
2/(x-1) - 1/2 = 4/(x^2-1)
Found 2 solutions by n2, ikleyn:
Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.
Solve:
- =
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Your starting equation is - = Its domain is the set of all real numbers except x = -1 and/or x = 1, where the denominator is zero. So, we look for solutions in the domain, where x =/= -1, x =/= 1. Multiply equation by = 2*(x-1)*(x+1). You will get 2*2*(x+1) - 4*2 = x^2 - 1, 4x + 4 - 8 = x^2 - 1, x^2 - 4x + 3 = 0, (x-3)*(x-1) = 0. The roots of this equation are x=3 and x = 1. But x=1 is not in the domain of the original equation, so we reject it. The only solution to the original equation is x = 3. ANSWER

Solved.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Solve:
- =
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The solution in the post by @mananth is incorrect.

@mananth found two potential solutions, x = 3 OR x=1, and presented them as a final solution.

                It is the ERROR.

In reality, only ONE of these two potential solutions is a real solution: it is x = 3.

x=1 is not in the domain of the equation, and, therefore, should be rejected.

It is a standard check to reject extraneous solutions, but @mananth neglects to make this check.

The correct answer is x = 3, the only one single unique solution.

Question 1210574: How would you graph 4x+3y=-24 show it to me in a graph
Found 3 solutions by ikleyn, greenestamps, josgarithmetic:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
How would you graph 4x+3y=-24 show it to me in a graph
~~~~~~~~~~~~~~~~~~~~~~

This equation is linear, so, it represents a straight line.

To graph a straight line, it is enough to have two points on a coordinate plane, or on a graph paper.

A simplest way to get two points is as follows.

(1) Plug x = 0 into equation. You will get 3y = -24, which gives you y = -8. So, one point is A = (x,y) = (0,-8). You can place it in a graph paper. (2) Next, plug y = 0 into equation. You will get 4x = -24, which gives you x = -6. So, another point is B = (x,y) = (-6,0). You can place it in a graph paper. (3) Having two points, A and B in a graph paper, you take a straightedge and a pencil and draw straight line through points A and B.

It is the way, how the students made this job 100, and 200, and 300 years ago.

In our days, you can use the advantages of technology.

Go to website https:\\www.desmos.com/calculator

Find there free of charge plotting tool for common use.

Print your equation there and get the plot in the next instance.

Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

As the other tutor says, you COULD put the equation in slope-intercept form and make the graph using the slope and y-intercept.

However, when the given equation is in this form, finding the equation in slope-intercept form requires extra algebra work.

Instead, use the x- and y-intercepts.

OOPS!!! I didn't see the "-" on the right side of the equation....

Here is my corrected response.

The x-intercept is the x value when y is 0:

The x-intercept is (-6,0)

The y-intercept is the value of y when x is 0:

The y-intercept is (0,-8)

Two points on the graph are (-6,0) and (0,-8). The graph is the straight line containing those two points.

Note that drawing the graph by using two known points is easier than by using the slope and y-intercept. So this method of drawing the graph requires no extra algebra, and the actual drawing of the graph is easier.

Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
You could put the equation into slope-intercept form, and plainly read the slope and vertical axis intercept from the equation to make the graph.

Slope is and a point on the line is (0, -8).

Question 1181614: What is the solution to the system?
-x-2y=2
2x+3y=-3

Found 3 solutions by n2, CPhill, ikleyn:
Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.
What is the solution to the system?
-x-2y=2
2x+3y=-3
~~~~~~~~~~~~~~~~~~~~~~~~~~~

@CPhill copy-pasted the solution by @mananth and placed it under his own name (even without acknowledgment).

Both "solutions" by @CPhill and by @mananth are identical and both are WRONG,
since they do not satisfy the original equations.

For correct solution, see the post by @ikleyn at this spot.

Ignore both posts by @CPhill and @mananth - their solutions both are irrelevant to the given problem.

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
-x-2y=2 -----------------1
2x+3y=-3------------------2
multiply (1) by 2
we get
-2x-4y=-4 Add this to eq (2)
2x+3y=-3
-y =-7
y=7
plug y in eq(1)
-x -2*7=2
-x= 16
x= -16
solution ( -16,7)

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
What is the solution to the system?
-x-2y=2
2x+3y=-3
~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth is INCORRECT.
        Easy check shows that his answer (-16,7) does not satisfy first equation.
        I came to bring a correct solution.

Your starting equations are -x - 2y = 2 (1) 2x + 3y = -3 (2) Multiply equation (1) by 2 and add with the second equation. You will get -y = 1, y = -1. Then from equation (1), x = -2y - 2 = -2*(-1) - 2 = 2 - 2 = 0. ANSWER. There is a unique solution x = 0, y = -1. CHECK. You may check it on your own, that my answer satisfies both given equations.

Solved correctly.

Question 1181419: Find the standard form of the equation of the line that passes through
(1/2, 2) and has slope -4.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Slope m=-4.00
Plug value of the slope and point ( 0.50 , 2 ) in
Y = m x + b
2.00 = -2 + b
b= 2.00 - -2
b= 4
So the equation will be
Y = -4 x + 4

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Find the standard form of the equation of the line that passes through
(1/2, 2) and has slope -4.
~~~~~~~~~~~~~~~~~~~~~~~~~~

Strictly speaking, the answer in the post by @mananth is not correct.

His equation y = -4x + 4 correctly describes the line, but his equation is slope-intercept form,

not the standard form.

Standard form equation is 4x + y = 4 in this problem.

Question 1210539: y= - 5
2
x+1
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
y= - 5
2
x+1
~~~~~~~~~~~~~~~~~~~~~~~~

        ? ? ? ? ? ? ?

Question 452212: solve the system by graphing:
y=-x-2
y=2x+13
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
Identify and use slope and y-intercept of each line and use them to graph the lines.

Read the graph's intersection point.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
Graphs/452212: solve the system by graphing:
y=-x-2
y=2x+13
~~~~~~~~~~~~~~~~~~~~~~~~~~

        My vision is that the instruction "solve the system by graphing" looks slightly idiotic
        as considered to this problem. So, I will solve it in a normal way, i.e. algebraically.

        Notice that the post b @mananth does not contain any solution.

Left sides of equations are identical - so, their right side equal -x - 2 = 2x + 13. Simply and find x -x - 2x = 13 + 2, -3x = 15, x = 15/(-3) = -5. Hence, y = -x - 2 = -(-5) - 2 = 5 - 2 = 3. ANSWER. The solution is x = -5, y = 3. You may substitute these values into equations to assure that they are the correct solutions, i.e. do satisfy equations.

Solved algebraically, i.e. in a normal way.

Question 440722: Could someone please help me to graph the line containing the given pair of points and find the slope?
(6, 3) (-5, -2)
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
points (6,3) and (-5,-2)

slope

Some point expected on the line, (x,y)

Now just use slope and axis intercept (0, 3/11) to
make the graph.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Could someone please help me to graph the line containing the given pair of points and find the slope?
(6, 3) (-5, -2)
~~~~~~~~~~~~~~~~~~~~~~~~~

        The exact solution (the required equation) to the problem should be expressed in rational numbers.
        But @mananth in his post works with decimal numbers with two decimals after the decimal point.
        Therefore, his equation IS NOT EXACT. Due to this reason, his solution can not be accepted
        as an ideal perfect solution to the problem.

        Below I develop such an ideal perfect EXACT equation.

Calculate the slope m = = = = 0.454545... So, an equation of the line in slope-intercept form is y = mx + b = + b. We should determine the value of 'b' in this equation. For it, substitute coordinates of the either given point into the equation 3 = + b, b = 3 - = = . The exact equation is y = . At this point, the problem solved PRECISELY and CORRECTLY.

Solved.

--------------------------

I know (I just deciphered it out for myself some time ago) that solutions by @matanth are produced by the computer code.

This mistake in the @manath solution (= in his code) should be fixed.

Question 435031: $2000 total deposit with 2 savings accounts. One pays interest rate of 6%the other at rate of 8% if a total earned interest is $144 how much is each deposit
Found 3 solutions by greenestamps, josgarithmetic, ikleyn:
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

This is essentially a mixture problem -- we are mixing money invested with a return of 6% and money invested at 8%.

Here is a solution using a method that can be used in any 2-part mixture problem like this.

All $2000 invested at 6% would yield a return of $120; all invested at 8% would yield a return of $160; the actual return is $144.

The ratio in which the money is invested in the two places is exactly determined by where the actual return of $144 lies between $120 and $160.

Use a number line if it helps to determine that $144 is three-fifths of the way from $120 to $160.

(The difference between $160 and $120 is $40; the difference between $144 and $120 is $24. $24 is 24/40 = 6/10 = 3/5 of $40)

That means 3/5 of the total $2000 was invested at the higher rate.

3/5 of $2000 is $1200, so $1200 of the total $2000 was invested at 8%.

ANSWERS: $1200 was invested at 8%; $800 at 6%.

CHECK: .08(1200)+.06(800) = 96+48 = 144

Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
How much time invested? How is the compounding?
Problem description is not complete.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
$2000 total deposit with 2 savings accounts. One pays interest rate of 6%,
the other at rate of 8% if a total earned interest is $144 how much is each deposit
~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution and the answer in the post by @mananth both are incorrect.
        His solution has arithmetic errors and does not withstand the standard check.
        So, I came to bring a correct solution.
        I will solve the problem using one single unknown, since it provides a shorter way.

Let x be the amount invested at 8%, in dollars. Then the amount invested at 6% is (2000-x) dollars. Write the total annual interest equation 0.08*(x + 0.06(2000-x) = 144 dollars. Simplify and find x 0.08x + 120 - 0.06x = 144, 0.08x - 0.06x = 144 - 120, 0.02x = 24 x = 24/0.02 = 2400/2 = 1200. ANSWER. $1200 invested at 8%; $the rest, $2000 - $1200 = $800 invested at 6%. CHECK. 0.08*1200 + 0.06*800 = 144 dollars. PRECISELY as it is given in the problem.

Solved correctly.

-------------------------------

This problem can be solved algebraically using one equation in one unknown or two equations in two unknowns.
It also can be solved mentally.

Since this site is to teach algebra, I place here algebraic solution.
Since the solution with one equation is simpler and shorter, I prefer this form with one unknown, which I present here.

What is really important, is to check the final solution for validity.

If you do not check, nothing will prevent you from mistakes.

Question 430837: x^2+8x+15>=0
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
x^2 + 8x + 15 >= 0
~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution and the answer in the post by @mananth both are incorrect.
        I came to bring a correct solution.

This request is to solve inequality x^2 + 8x + 15 >= 0. Left side can be factored (x+5)*(x+3) >= 0. The solution for this inequality is the union of two sets of real numbers x <= -5 OR x >= -3. In the interval form, the solution is (,] U [,).

Solved correctly.

Question 1165384: The graph of f(x) is shown below. https://latex.artofproblemsolving.com/d/b/a/dba24dce7c2734af95b18f61b73b2a32f213cd79.png
For each point (a,b) on the graph of y = f(x), the point ( 3a - 1, b/2) is plotted to form the graph of another function y = g(x). For example, (0,2) lies on the graph of y = f(x), so (3 * 0 - 1, 2/2) = (-1,1) lies on the graph of y = g(x).
(a) Plot the graph of y = g(x). Include the diagram in your solution.
(b) Express g(x) in terms of f(x).
(c) Describe the transformations that you would apply to the graph of y = f(x) to obtain the graph of y = g(x). For example, one transformation might be to stretch the graph horizontally by a factor of 5.
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
This problem involves understanding function transformations based on how the coordinates of points change.
Since the graph of $f(x)$ is not provided, I will define a simple, representative graph with clear points to illustrate the transformations.
## Example Graph: $y = f(x)$
Let's assume the graph of $f(x)$ has the following four distinct points, which define its shape :
* $P_1 = (-4, 4)$
* $P_2 = (0, -2)$
* $P_3 = (4, 2)$
---
## (a) Plot the graph of $y = g(x)$
The transformation rule is: Every point $(a, b)$ on the graph of $y = f(x)$ becomes the point $(5a, b - 3/2)$ on the graph of $y = g(x)$.
Applying the transformation $(x, y) \to \left(5x, y - \frac{3}{2}\right)$ to our example points:
* $P_1(-4, 4) \to P'_1 \left(5(-4), 4 - \frac{3}{2}\right) = P'_1 (-20, 2.5)$
* $P_2(0, -2) \to P'_2 \left(5(0), -2 - \frac{3}{2}\right) = P'_2 (0, -3.5)$
* $P_3(4, 2) \to P'_3 \left(5(4), 2 - \frac{3}{2}\right) = P'_3 (20, 0.5)$
The graph of $y = g(x)$ connects the points $(-20, 2.5)$, $(0, -3.5)$, and $(20, 0.5)$.

---
## (b) Express $g(x)$ in terms of $f(x)$
To find the function $y = g(x)$, we reverse the transformations applied to the coordinates.
Let $(X, Y)$ be a point on the graph of $g(x)$, and let $(x, y)$ be the corresponding point on the graph of $f(x)$.
The transformation rule is:
1. $X = 5x$
2. $Y = y - \frac{3}{2}$
We need to solve for $x$ and $y$ in terms of $X$ and $Y$:
1. $x = \frac{1}{5} X$
2. $y = Y + \frac{3}{2}$
Since $(x, y)$ is on the graph of $f(x)$, we have $y = f(x)$. Substitute the expressions for $x$ and $y$:
$$Y + \frac{3}{2} = f\left(\frac{1}{5} X\right)$$
Now, solve for $Y$:
$$Y = f\left(\frac{1}{5} X\right) - \frac{3}{2}$$
Replacing $Y$ with $g(X)$ and using $x$ as the standard variable:
$$\mathbf{g(x) = f\left(\frac{1}{5} x\right) - \frac{3}{2}}$$
---
## (c) Describe the transformations
The expression $g(x) = f\left(\frac{1}{5} x\right) - \frac{3}{2}$ reveals two transformations applied to $f(x)$:
1. **Horizontal Transformation (due to $\frac{1}{5}x$ inside $f$):**
The $x$-coordinates are multiplied by 5, which corresponds to a **horizontal stretch by a factor of 5**.
$$\left(x \to \frac{1}{5}x\right) \implies \text{Horizontal Stretch by } 5$$
2. **Vertical Transformation (due to $-\frac{3}{2}$ outside $f$):**
The constant $\frac{3}{2}$ is subtracted from the $y$-value, which corresponds to a **vertical shift downward by $\frac{3}{2}$ units (or $1.5$ units)**.
$$\left(y \to y - \frac{3}{2}\right) \implies \text{Vertical Shift Down by } \frac{3}{2}$$

Question 1165421: The graph of f(x) is shown below. https://latex.artofproblemsolving.com/d/b/a/dba24dce7c2734af95b18f61b73b2a32f213cd79.png
For each point (a,b) on the graph of y = f(x), the point ( 3a - 1, b/2) is plotted to form the graph of another function y = g(x). For example, (0,2) lies on the graph of y = f(x), so (3 * 0 - 1, 2/2) = (-1,1) lies on the graph of y = g(x).
(a) Plot the graph of y = g(x). Include the diagram in your solution.
(b) Express g(x) in terms of f(x).
(c) Describe the transformations that you would apply to the graph of y = f(x) to obtain the graph of y = g(x). For example, one transformation might be to stretch the graph horizontally by a factor of 5.
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
This problem involves understanding function transformations based on how the coordinates of points change.
Since the graph of $f(x)$ is not provided, I will define a simple, representative graph with clear points to illustrate the transformations.
## Example Graph: $y = f(x)$
Let's assume the graph of $f(x)$ has the following four distinct points, which define its shape :
* $P_1 = (-4, 4)$
* $P_2 = (0, -2)$
* $P_3 = (4, 2)$
---
## (a) Plot the graph of $y = g(x)$
The transformation rule is: Every point $(a, b)$ on the graph of $y = f(x)$ becomes the point $(5a, b - 3/2)$ on the graph of $y = g(x)$.
Applying the transformation $(x, y) \to \left(5x, y - \frac{3}{2}\right)$ to our example points:
* $P_1(-4, 4) \to P'_1 \left(5(-4), 4 - \frac{3}{2}\right) = P'_1 (-20, 2.5)$
* $P_2(0, -2) \to P'_2 \left(5(0), -2 - \frac{3}{2}\right) = P'_2 (0, -3.5)$
* $P_3(4, 2) \to P'_3 \left(5(4), 2 - \frac{3}{2}\right) = P'_3 (20, 0.5)$
The graph of $y = g(x)$ connects the points $(-20, 2.5)$, $(0, -3.5)$, and $(20, 0.5)$.

---
## (b) Express $g(x)$ in terms of $f(x)$
To find the function $y = g(x)$, we reverse the transformations applied to the coordinates.
Let $(X, Y)$ be a point on the graph of $g(x)$, and let $(x, y)$ be the corresponding point on the graph of $f(x)$.
The transformation rule is:
1. $X = 5x$
2. $Y = y - \frac{3}{2}$
We need to solve for $x$ and $y$ in terms of $X$ and $Y$:
1. $x = \frac{1}{5} X$
2. $y = Y + \frac{3}{2}$
Since $(x, y)$ is on the graph of $f(x)$, we have $y = f(x)$. Substitute the expressions for $x$ and $y$:
$$Y + \frac{3}{2} = f\left(\frac{1}{5} X\right)$$
Now, solve for $Y$:
$$Y = f\left(\frac{1}{5} X\right) - \frac{3}{2}$$
Replacing $Y$ with $g(X)$ and using $x$ as the standard variable:
$$\mathbf{g(x) = f\left(\frac{1}{5} x\right) - \frac{3}{2}}$$
---
## (c) Describe the transformations
The expression $g(x) = f\left(\frac{1}{5} x\right) - \frac{3}{2}$ reveals two transformations applied to $f(x)$:
1. **Horizontal Transformation (due to $\frac{1}{5}x$ inside $f$):**
The $x$-coordinates are multiplied by 5, which corresponds to a **horizontal stretch by a factor of 5**.
$$\left(x \to \frac{1}{5}x\right) \implies \text{Horizontal Stretch by } 5$$
2. **Vertical Transformation (due to $-\frac{3}{2}$ outside $f$):**
The constant $\frac{3}{2}$ is subtracted from the $y$-value, which corresponds to a **vertical shift downward by $\frac{3}{2}$ units (or $1.5$ units)**.
$$\left(y \to y - \frac{3}{2}\right) \implies \text{Vertical Shift Down by } \frac{3}{2}$$

Question 731836: Sketch a graph for IxI=4

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Sketch a graph for IxI=4
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This forum is not a plotting studio to serve such requests.

Go to website http://www.desmos.com/calculator/

Find there free of charge plotting tool for common use.

Print the formula for your function there.

Get the plot instantly.

Question 1210484: How would you graph y=-2/3+5 and -2x+4y=-8 with a point of (6,1)

Found 4 solutions by josgarithmetic, timofer, ikleyn, CPhill:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
Yes. Each of the equations contain the point (6,1).

Answer by timofer(155)   (Show Source):
You can put this solution on YOUR website!
The question is missing or wrong itself.
What do you have and what do you want to ask?

Is this, "how do you graph the system of equations y=-(2/3)x+5 and -2x+4y=-8 ?" And then how do you intend that point (6,1) to be used?

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
How would you graph y=-2/3+5 and -2x+4y=-8 with a point of (6,1)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

As the problem is formulated/presented in the post,
it is self-contradictory and has no solution.

It looks like first equation is written incorrectly.

===========================================

In his response, @josgarithmetic says "Yes", but his "Yes" does relate to another, different problem.

So, his response "Yes" goes via hidden "No", which he hesitates to pronounce.

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
It looks like you want to **graph two lines** and determine if the point $(6, 1)$ lies on their intersection.
The first equation, $y = -2/3 + 5$, is simply a horizontal line since it's a constant value. The second equation, $-2x + 4y = -8$, is a linear equation in standard form.
Let's first clarify and re-write the equations for graphing.
## 1. Analyze and Simplify the Equations
### Equation 1: $y = -2/3 + 5$
This equation simplifies to a single constant:
$$y = -\frac{2}{3} + \frac{15}{3}$$
$$y = \frac{13}{3} \approx 4.33$$
This is a **horizontal line** where every point has a y-coordinate of $\frac{13}{3}$.
### Equation 2: $-2x + 4y = -8$
It's easiest to graph this line using the slope-intercept form ($y = mx + b$).
1. **Isolate $y$:**
$$4y = 2x - 8$$
2. **Divide by 4:**
$$y = \frac{2}{4}x - \frac{8}{4}$$
$$y = \frac{1}{2}x - 2$$
This line has a **y-intercept ($b$) of $-2$** and a **slope ($m$) of $\frac{1}{2}$** (up 1, right 2).
---
## 2. Graphing the Lines
To graph these lines, you'll plot points and draw the lines on a coordinate plane.
### Line 1: $y = \frac{13}{3}$
1. **Plot the y-intercept:** $(\mathbf{0, \frac{13}{3}})$ or $(0, 4.33)$.
2. **Draw a horizontal line** passing through that point.
* Example points: $(3, \frac{13}{3})$, $(-3, \frac{13}{3})$.
### Line 2: $y = \frac{1}{2}x - 2$
1. **Plot the y-intercept:** $(\mathbf{0, -2})$.
2. **Use the slope** ($m = \frac{1}{2}$) to find other points:
* From $(0, -2)$, go up 1 and right 2 to find $(\mathbf{2, -1})$.
* From $(2, -1)$, go up 1 and right 2 to find $(\mathbf{4, 0})$.
3. **Draw a straight line** through these points.
---
## 3. Check the Given Point $(6, 1)$
Finally, let's see if the point $(6, 1)$ lies on **either** line.
### Check Line 1: $y = \frac{13}{3}$
Substitute $y=1$:
$$1 = \frac{13}{3}$$
This is **False**. The point $(6, 1)$ is **not** on Line 1.
### Check Line 2: $y = \frac{1}{2}x - 2$
Substitute $x=6$ and $y=1$:
$$1 = \frac{1}{2}(6) - 2$$
$$1 = 3 - 2$$
$$1 = 1$$
This is **True**. The point $(6, 1)$ **is** on Line 2.
The point $(6, 1)$ is only on the line $y = \frac{1}{2}x - 2$.
Would you like to find the exact point where these two lines intersect?

Question 1210482: y-8=-x
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
```python?code_reference&code_event_index=2
import matplotlib.pyplot as plt
import numpy as np
# Define the function
def line_function(x):
return -x + 8
# Generate x values
x_values = np.linspace(-5, 10, 400)
# Generate y values
y_values = line_function(x_values)
# Create the plot
plt.figure(figsize=(8, 6))
plt.plot(x_values, y_values, label='$y = -x + 8$')
# Highlight the intercepts
plt.plot(0, 8, 'ro', label='y-intercept (0, 8)')
plt.plot(8, 0, 'go', label='x-intercept (8, 0)')
# Add labels and title
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.title('Graph of $y - 8 = -x$')
# Set grid and axes
plt.axhline(0, color='black', linewidth=0.5)
plt.axvline(0, color='black', linewidth=0.5)
plt.grid(True, linestyle='--', alpha=0.6)
# Set limits to clearly show the intercepts
plt.xlim(-2, 10)
plt.ylim(-2, 10)
# Add legend
plt.legend()
# Save the plot
plt.savefig('graph_y_minus_8_equals_minus_x.png')
print("graph_y_minus_8_equals_minus_x.png")
```
```text?code_stdout&code_event_index=2
graph_y_minus_8_equals_minus_x.png
```
[image-tag: code-generated-image-0-1764864400960668627]
The equation $y - 8 = -x$ can be easily graphed by rewriting it in the **slope-intercept form** ($y = mx + b$).
$$y = -x + 8$$
This line has the following key properties:
* **Slope ($m$):** $\mathbf{-1}$ (or $\frac{-1}{1}$), meaning for every $1$ unit you move right, you move $1$ unit down.
* **Y-intercept ($b$):** $\mathbf{8}$, meaning the line crosses the $y$-axis at the point $(\mathbf{0, 8})$.
* **X-intercept:** $\mathbf{8}$, meaning the line crosses the $x$-axis at the point $(\mathbf{8, 0})$.
The graph below shows the line $y = -x + 8$.

Question 733209: Graph the linear equation:
2x+5y+24=0
~What I tried~
2x+5y+24=0
-24 -24
-------------
2x+5y=-24
-2x -2x
-------------
5y=-24-2x
/5 /5
-------------
y=-4.8-2/5x
Swap these^ around.
y=-2/5x-4.8
But I'm not sure if this is correct.
Thanks.
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

Your steps are correct.

My congratulations !

It is a real success !

Question 481534: I'm struggling with this. I need to find the vertex, line of symmetry, and the max or min value of f(x).
The equation is f(x)=1/3 (x+7)^2 + 9
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
I'm struggling with this. I need to find the vertex, line of symmetry, and the max or min value of f(x).
The equation is f(x)=1/3 (x+7)^2 + 9
~~~~~~~~~~~~~~~~~~~~~~~~~

        In his post, tutor @Theo makes a lot of unnecessary calculations.
        If you will follow his solution, your teacher will see that you do not understand the basic conceptions
        of the subject.

        The problem can be and should be solved in couple of lines.
        Do not follow the @Theo' solution, since it is wrong way teaching.

You are given an equation of a parabola in the vertex form. Having this equation, you immediately see that the line of symmetry is x=-7 and the vertex point is (-7,9). Minimum value of the function is 9 at x = -7 (at the vertex point).

Solved.

---------------------------

This is all that your teacher wants to get from you.

I don't know, for what reason tutor @Theo to bulk up tons of unnecessary calculations in many
of his posts. It is definitely wrong style teaching --- I would even say - unacceptable style of teaching.

It is why I re-write one his "solution" after another.

Question 490209: this is first time algebra and it is confusing. were do i start with this proble
profit=$0.15 x number of e-readers sold - $28
values given in 1000
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
this is first time algebra and it is confusing. were do i start with this proble
profit=$0.15 x number of e-readers sold - $28
values given in 1000
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

To me, neither the post formulation, nor the solution by @Theo are clear.

I would not trust neither first nor second.

To the reader: if you happen to read this problem and the solution by @Theo,
my advise to you is to ignore both of them and do not spend your time for nothing.

Question 551377: 2000 students in the whole school and only 55 of 90 students are buying yearbooks approximately how many books should be orderd?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
2000 students in the whole school and only 55 of 90 students are buying yearbooks approximately.
How many books should be ordered ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        Tutor @Theo solved this problem in his post, but I came to present my solution
        and some additional thoughts about the rounding of final number.

Similar as @Theo did in his post, you apply the ratio of 55/90 to 2000 to get 55/90 * 2000 = 1222.2222 students requiring books. Formal rule says to round to 1222, but the common sense says to round to 1223. Which way will you prefer ? My common sense tells me to round to 1223. Potentially, there are 1222.222 students, who want to buy yearbooks, and if the school wants to serve everyone of potential buyers, the school should have 1223 textbooks in the store.

This problem illustrates that in some cases common sense OVERLAYS the formal rules.

This statement can be re-phrased in other words: some problems DICTATE to round
differently than the formal rules.

It is worth to learn, and this problem is a good example.

-----------------------------

In this problem, a reasonable mathematical answer says that, based on given data,
the school should order at least 1223 yearbooks.

////////////////////////////////////////////

Interesting fact:

Today, Oct.14, 2025, I submitted this problem to Google AI (" Overview AI ").

It was interesting to me, how this AI will treat this problem.

It created the answer exactly as in the post by @Theo.

It is very natural, since Google AI is not able to think:   it is programmed
to rewrite from its database solutions, and it is all what it really can do in Math problems.

Naturally, 1222 is the lame answer.

I reported to them via their feedback system.
Hope, in the feature, Google AI will be more smart, having my solution in its database.

It is why I am checking the solutions by other tutors and fix/correct them, when required.
Otherwise, thousands of students will have millions defective solutions from Artificial Intelligence,
so, their mind and their common sense will be injured by the AI,
while this AI will be beaten by the competitors.

I am a devoted partner of artificial intelligence, and I work tirelessly
to ensure that the solutions to Math problems in its database are correct.

Question 1166975: find the Range of f(x) = (5/(x + 1)) cos (x) , x ≥ 0
Found 2 solutions by ikleyn, amarjeeth123:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
find the Range of f(x) = (5/(x + 1))*cos (x) , x ≥ 0.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I will show here two ways to find the range. First way is to use plotting tool like DESMOS, which you can find at the site https://www.desmos.com/calculator/ Print there the formula of the function, and you will get the plot immediately. Click on the plot somewhere around the first minimum of the function in the domain x >= 0. The plotting tool will show you the coordinates of the minimum. They are (x,y) = (2.88997,-1.24488). So, the minimum value of the function in the domain x >= 0 is -1.24488 (approximately). Thus the range is [ , ). I saved the plot under this link https://www.desmos.com/calculator/9x4t38gmur for documentation purposes, so you can see it at any time. To see the coordinates of the minimum of the function, click on the plot in vicinity of the minimum. Another way to find the range is to find the minimum of the function in the domain x >= 0 using Calculus. Take the derivative of the function and equate it to zero. You will get an equation tan(x) = . It can not be solved analytically, but can be solved numerically. Use an online calculator for solving transcendent equations https://www.wolframalpha.com Print there this equation tan(x) = . The calculator will get you a series of the roots. Your root is the first positive number in this series x = 2.88997 (approximately). To get the minimum of the function , substitute x = 2.88997 into the function. You will get the value -1.24488 for this minimum. So, the range of the function is [ , ), which is the same as we found it in the first solution.

Thus the problem solved completely in two ways for your better understanding.

Answer by amarjeeth123(574)   (Show Source):
You can put this solution on YOUR website!
We cannot determine the range of the function.
The domain is {x element R : x!=-1}

Question 1165235: Suppose that the function f is defined for all real numbers as follows.
f(x)= -x^(2)+10 if -4<=x<4
-5-x if x>=4
Graph the function f. Then determine whether or not the function is continuous.
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

This is the graph of the piecewise function.

The red parabola is due to the piece y = -x^2+10, when graphed on the interval -4 <= x < 4.
Graph y = -x^2+10 as normal, then erase the portions when x < -4 or x > 4.
The vertex is at (0,10).
Two other points on this parabola are (-1,9) and (1,9).

Two graphing tools I recommend are Desmos and GeoGebra
Or you can use something like a TI83.
Another alternative is to use graph paper to do everything by hand.

The blue line is the piece y = -5-x on the interval x >= 4.
Two points on this blue line are (4,-9) and (5,-10).

There is a red closed endpoint at (-4,-6)
There is an open hole at (4,-6) marked in red.
There is a blue closed filled in endpoint at (4,-9)
Another way of saying "closed endpoint" could be "filled in endpoint".

The open endpoint is due to the lack of "or equal to" in the inequality sign.
This is why we exclude x = 4 from the red parabola.
In contrast every other endpoint is included because there is an "or equal to".

We can clearly see the graph is not continuous.
There's a jump at x = 4 when going from the red curve to the blue line.

--------------------------------------------------------------------------

To determine this algebraically, without needing a graph, plug x = 4 into both pieces to see what results.
I'll label the pieces g(x) and h(x).
g(x) = -x^2+10
g(4) = -4^2+10
g(4) = -16+10
g(4) = -6
This leads to the location (4,-6) which is the red open hole on the graph.

And,
h(x) = -5-x
h(4) = -5-4
h(4) = -9
This leads to the blue filled in endpoint at (4,-9)
The results -6 and -9 do not agree on the same number, so this is a non-graph confirmation that the piecewise function is not continuous.
To be continuous, we would need g(4) = h(4) to be the case.

Similar practice problems are at the following links
https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1206484.html
https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1188881.html
https://www.algebra.com/algebra/homework/Rational-functions/Rational-functions.faq.question.1200349.html

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Suppose that the function f is defined for all real numbers as follows.
f(x)= -x^(2)+10 if -4<=x<4
-5-x if x>=4
Graph the function f. Then determine whether or not the function is continuous.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In each of the two domains, the function is defined as a polynomial. Therefore, inside each domain the function is continuous (since a polynomial is a continuous function, as you should know from Calculus). Hence, to be a continuous function over the entire domain, the necessary and sufficient condition for the given function is to have the limit at x= 4 from the left to be equal to its value at x = 4. The limit at x = 4 from the left side is -x^2 + 10 at x ---> 4, i.e. -4^2+10 = -16+10 = -6. The value of the function at x = 4 is -5 - 4 = -9. -9 =/= -6, so we conclude that the given function is not continuous at x = 4.

At this point, we complete our reasoning and answered the question.

//////////////////////////////////////

Notice that the problem's formulation has a deficiency.

Indeed, it says that "the function f is defined for all real numbers as follows . . . ",
but in reality it is defined only on the union of two intervals [-4,4) and [4,).

This union is the whole interval [,), but it is not the set of all real numbers,
as the problem proclaims. Thus, the problems wording is not perfect and is not accurate.

By such signs, every professional Math reader always can unmistakably recognize
"problems" that were written by an unprofessional "in a garage on the knee."

By the way, the first three words in the problem "Suppose that the" are excessive.
They are not necessary and can be omitted. Then the problem's appearance will be better.

It is a rule of Math writing: remove everything that does not matter.

In Math, following this rule is expressing of respect to a reader.

Question 1210399: The Piecewise function of this
f(x) {-x if x <2}
{x^2-6x+9 if x>2 }
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

On your request, I prepared a plot for this given piece-wise function.

See the link

https://www.desmos.com/calculator/x0xmwwlmtj
https://www.desmos.com/calculator/x0xmwwlmtj

I used online plotting tool DESMOS at this site

www.desmos.com/calculator/

which is free of charge for common use.

To get the plot, you simply print the formulas for the function.

The only trick is to write the condition "if" .
For this purpose, you should use curved brackets.
See how I did it in the referred plot.

Question 1206470: find the slope of a line that is perpendicular to the y-axis and passes through the point(-7,8)
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
find the slope of a line that is perpendicular to the y-axis and passes through the point(-7,8)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

To answer this question, you should not make any calculations, and the given coordinates
of the point are excessive unnecessary information.

Any straight line in a coordinate plane, perpendicular to y-axis, is parallel to x-axis and has zero slope.

That is all.

Question 1210357:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

Empty post.

Question 1209656: What is the smallest distance between the origin and a point on the graph of y = \frac{1}{\sqrt{3}} (x^2 - 7 + 2x)?
Found 2 solutions by mccravyedwin, ikleyn:
Answer by mccravyedwin(421)   (Show Source):
You can put this solution on YOUR website!

On your TI-84 graphing calculator Press y= Enter √(x²+(1/3)(x²-7+2x)²) Press zoom 6 See this graph Press 2nd trace Choose 3:minimum Use arrow key to place cursor a little left of the minimum point. Press enter Use arrow key to place cursor a little right of the minimum point. Press enter twice See cursor move to minimum point Read at the bottom of screen X=1.6578865 Y=1.7436727 So the mimimum value of y is 1.7436727, approximately. Edwin

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
What is the smallest distance between the origin and a point on the graph of y = ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The square of the distance from the origin to the point (x,y) is D = x^2 + y^2. We have D = x^2 + y^2 = x^2 + I will not simplify this expression. Instead, I will find the minimum of D over {x} graphically. I use the plotting tool at website www.desmos.com/calculator/ It provides the plot (free of charge) and the position and the coordinates of the minimum. My plot is under this link https://www.desmos.com/calculator/4phmfmwfj7 To see the coordinates of the minimum, click on it. We have = 3.04039 approximately. Hence, the minimum distance under the problem's question is = 1.7437. ANSWER. The smallest distance between the origin and a point on the graph of y = is 1.7437, approximately.

Solved.

Question 1210255: find the x intercept and the y intercept. then use them to graph the line. 9x6y=27
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
You possibly really have the equation 9x+6y=27, or maybe 9x-6y=27.

Question 1210143: Let R be the image of rotating point P=(4,0) counterclockwise by 60^\circ degrees around Q=(12,-7). What is PR?
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Let $P = (4, 0)$ and $Q = (12, -7)$.
We want to rotate $P$ counterclockwise by $60^\circ$ around $Q$ to obtain point $R$.
We want to find the distance $PR$.
First, let's find the vector $\vec{QP} = P - Q = (4-12, 0-(-7)) = (-8, 7)$.
Let's rotate this vector counterclockwise by $60^\circ$.
We can represent the vector $\vec{QP}$ as a complex number: $z = -8 + 7i$.
To rotate $z$ counterclockwise by $60^\circ$, we multiply it by $e^{i\pi/3} = \cos(60^\circ) + i\sin(60^\circ) = \frac{1}{2} + i\frac{\sqrt{3}}{2}$.
The rotated vector is:
$$z' = z \cdot e^{i\pi/3} = (-8 + 7i)\left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) = -4 - 4i\sqrt{3} + \frac{7i}{2} - \frac{7\sqrt{3}}{2} = \left(-4 - \frac{7\sqrt{3}}{2}\right) + i\left(\frac{7}{2} - 4\sqrt{3}\right)$$
This corresponds to the vector $\vec{QR} = \left(-4 - \frac{7\sqrt{3}}{2}, \frac{7}{2} - 4\sqrt{3}\right)$.
So, $R = Q + \vec{QR} = \left(12 - 4 - \frac{7\sqrt{3}}{2}, -7 + \frac{7}{2} - 4\sqrt{3}\right) = \left(8 - \frac{7\sqrt{3}}{2}, -\frac{7}{2} - 4\sqrt{3}\right)$.
We want to find $PR$.
Let's first find $PQ = \sqrt{(12-4)^2 + (-7-0)^2} = \sqrt{8^2 + 7^2} = \sqrt{64+49} = \sqrt{113}$.
Since rotation preserves distances, $QR = PQ = \sqrt{113}$.
We want $PR$, not $QR$.
Since we are rotating $P$ around $Q$, the distance $QP$ is the same as $QR$.
We want to find the distance between $P$ and $R$.
We know $QP = QR = \sqrt{113}$.
Let $PR = d$.
We have a triangle $PQR$ with $QP = QR = \sqrt{113}$ and $\angle PQR = 60^\circ$.
Since $QP = QR$, $\triangle PQR$ is an isosceles triangle.
Since $\angle PQR = 60^\circ$, the other two angles are equal, and their sum is $180^\circ - 60^\circ = 120^\circ$.
Thus, $\angle QPR = \angle QRP = 60^\circ$.
Therefore, $\triangle PQR$ is an equilateral triangle, so $PR = QP = QR = \sqrt{113}$.
Final Answer: The final answer is $\boxed{\sqrt{113}}$

Question 1170320: If the room has a length of 12m more than its width and an area of more than or equal to 108m², how many tiles to cover completely the room considering that a tile has an area of less than or equal to 1600cm²? What is my preferred budget if one tile cost more or less 56 pesos(Philippine money)?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
If the room has a length of 12m more than its width and an area of more than or equal to 108m²,
how many tiles to cover completely the room considering that a tile has an area of less than or equal to 1600cm²?
What is my preferred budget if one tile cost more or less 56 pesos(Philippine money)?
~~~~~~~~~~~~~~~~~~~~~~~~~~

The problem is posed incorrectly.

As it is posed, it can not be answered in a reasonable way.

As the problem is posed, it demonstrates a total absence
of common sense on the side of its creator/composer.

My condolences . . .

Question 1209884: What is the horizontal asymptote as x approaches positive infinity of the graph of
y = \sqrt{4x^2 + 5x} - \sqrt{4x^2}?
The horizontal asymptote is in the form y = mx + k.

Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

!!!! The equation of a horizontal asymptote is not in the form y = mx + k, unless you are letting m be 0. The equation of a horizontal asymptote is of the form y = k.

Ignoring that (or allowing the slope m to be 0)....

Rationalize the numerator:

As x goes to positive infinity, goes to 0 so goes to = 1, and the expression approaches

ANSWER:

(or ....)

A graph....