Questions on Algebra: Graphs, graphing equations and inequalities answered by real tutors!

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First, let's complete the square.
f(x) = =
=
=
=
This tells us that the vertex of the parabola is (-3/4, -17/8), which is backwards from what you stated in the problem. When you say that you are stuck figuring out the other two points, I assume you are referencing the x and y intercepts. The y intercept is obtained by setting x to 0 in the original problem. So f(0) = -1 means the y intercept is (0, -1). In order to find the x intercepts, you set y = 0, and this is easier using the form above (found by completing the square).
So
0 =
17/8 =
17/16 =

So , which gives you your two x intercepts. Hope this helps!

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When you are dealing with inequalities then you cannot be dealing with equalities, right? The word inequality means NOT equal. The solution in terms of an inequality is what makes the inequality a true statement.
SAMPLE: Solve for x.
x + 5 < 10
We subtract 5 from both sides and get x < 5.
What does that mean?
It means that whatever the number for x is, it MUST BE LESS THAN 5 in order to make the original inequality true.
THE PROVE:
I found that x < 5.
TEST 1
Let x be GREATER than 5.
I will use 20.
So, x = 20.
I plug that into x + 5 < 10
20 + 5 < 10
25 < 10...Is this true? Is 25 LESS THAN 10?
This is a false statement, right?
Test 2
Let x be LESS THAN 5.
I will use 3.
So, x = 3.
x + 5 < 10
3 + 5 < 10
8 < 10...Is this true?
Yes, 8 is less than 10.
This is the basic idea regarding all inequality questions.
Understand?

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What's the equation of a line that passes through points (0,-1) and (2, 3 )?
---------------------
The easiest way to do these is using determinants.
------------
|x y 1|
|0 -1 1| = 0
|2,3 1|
--------
x*(-1-3) - y*(0-2) + 0+2 = 0
-4x + 2y = -2
2x - y = 1
---------
Both points fit.

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Start with the given inequality



Factor the left side


Set the left side equal to zero


Set each individual factor equal to zero:

or

Solve for x in each case:

or


So our critical values are and

Now set up a number line and plot the critical values on the number line





So let's pick some test points that are near the critical values and evaluate them.


Let's pick a test value that is less than (notice how it's to the left of the leftmost endpoint):

So let's pick

Start with the given inequality


Plug in


Evaluate and simplify the left side

Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.


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Let's pick a test value that is in between and :

So let's pick

Start with the given inequality


Plug in


Evaluate and simplify the left side

Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.
So part our solution in interval notation is ()





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Let's pick a test value that is greater than (notice how it's to the right of the rightmost endpoint):

So let's pick

Start with the given inequality


Plug in


Evaluate and simplify the left side

Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.


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





Answer:

So the solution is


So the solution in interval notation is:


()

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15x + 20y = 400
What is the process to solve this equation for x and y?
---------------
It's not possible. When you have 2 unknowns, you need 2 equations.

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parallel because slope = for both lines

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write equation in slope intercept form...y=mx+b
:
y=-3x+5
:
3y=x+6--->y=(1/3)x+2
:
now compare the slopes one is -3 and one is 1/3.....those are negative recipricals of each other therefore these lines are perpendicular to one another

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The formula for the slope is m=(y2-y1)/(x2-x1)
X
Substituting the given values for the two points we have
m=(-4-0)/(0-3)= -4/-3=4/3
The point-slope formula for a line is (y-y1)=m(x-x1)
So substituting the value for m and the values for x1 and y1 we have
(y-0)=4/3*(x-3)
Carrying out the arithmetic we have y = 4/3*(x-3)
I hope that this helps

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First let's find the slope of the line through the points and


Start with the slope formula.


Plug in , , , and


Subtract from to get


Subtract from to get


Reduce


So the slope of the line that goes through the points and is


Now let's use the point slope formula:


Start with the point slope formula


Plug in , , and


Simplify


Distribute


Multiply



So the equation that goes through the points and is


Notice how the graph of goes through the points and . So this visually verifies our answer.
Graph of through the points and

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what is the slope intercept form of the equation with the points (3,0) and (0,-4)?
--------------
|x y +1|
|3 0 +1| = 0
|0 -4 1|
-----------------
x*(0+4) - y*(3-0) -12 = 0
4x -3y = 12
3y = 4x - 12
y = (4/3)x -4

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what is the rise and the run of a line in the cartesian plane that passes through the points (-2,4) and (2,-1)?
--------------------
rise = change in "y" = (4--1) = 5
----
corresponding run is change in "x" = (-2-2) = -4
==================================================
Cheers,
Stan H.

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what is the equation of the line that passes through the points (-2,2) and (0,5)?
--------
|x y 1|
|-2 2 1| = 0
|0 5 1|
x*(2-5) - y*(-2) + -10 = 0
-3x + 2y -10 = 0
3x - 2y = -10

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what is the equation of the line that passes through the points (-2,2) and (0,5)?
----------------
slope = (5-2)/(0--2) = 3/2
--------------
intercept: (0,5)
---------------
EQUATION:
y = (3/2)x + 5
====================
Cheers,
Stan H.

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Since two points are given, you first need to find the slope (m) of the line.
This is the general formula for the slope.
m=(y2-y1)/(x2-x1) Substituting the actual values into the formula we have
m = (5-2)/(0-(-2))= 3/2
Now you can use the point-slope formula
(y-2)=3/2*(x-(-2))
So y = 3/2x+3+2= (3/2)x+5
As a check, substitute the value for x=0 for the other given point.
y=(3/2)*0+5=5 (Checks out OK)
I hope that this helps

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you can get infinite points from this line equation
:
choose a value for x or y and solve for the unknown variable
:
example: I choose 0 and 1 for x...so I will solve for y for those 2 values
:
x=0--->y=2(0)+3=3--->(0,3)
:
x=1--->y=2(1)+3=5--->(1,5)
:
now you have 2 points

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y=mx+b

First find the slope (m) using the difference of the y's over the difference of the x's.




so the slope is 1
y=x+b

Plug in one of the points, it doesn't matter which one.
0=4+b

Solve for b
b=-4

Answer:
y=x-4

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the rise is the diffence between the y coordinates and the run is the difference between the x coordinates (-2,4)=(x1,y1) (2,-1)=(x2,y2)
:
m=slope = rise/run==(y2-y1)/(x2-x1) or (y1-y2)/(x1-x2)
:
(4-(-1))/(-2-2)=-(5/4)

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When , that is the y-intercept. All you need to do
is find when the point on the line is (,)
To do that, plug into the equation




This is the y-intercept

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Please help me find the y-intercept of the line with the equation 4x+2y=12.
---------------
Let x = 0 then y = 6.
Intercept (0,6)
================
Cheers,
Stan H.

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to find the y intercept set x=0
:
4(0)+2y=12
:
2y=12
:

:
y intercept is (0,6)
you can also write this in slope intercept form which is y=mx+b where b is the slope intercept
:
2y=-2x+12
:
y=-x+6
:
therefore y intercept is (0,6)

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First write the equation of the line
In point slope form, the equation is
y-3=2*(x-(-1))
This can be written in slope y-intercept form
y=2*x+2+3 = 2*x+5
This line goes through the point(-1,3)
Select the "x" value for another point, say x=2.
Replace the variable "x" in the equation with the value of x which is 2.
Y= 2*2+5=9
So another point on the line is (2,9)

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you have to find the equation of the line then you can find any point. Use the point slope formula which states y-k=m(x-h) where m is the slope and (h,k) is any point on the line
:
y-3=2(x-(-1))
:
y-3=2x+2
:
y=2x+5
:
now just choose any value of x or y and solve for the unknown...I choose x=0
y=2(0)+5--->y=5
:
so for x=0,y=5--->point(0,5)

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If a line passes through the point (-1,3) and has a slop of 2. How do I find another point?
-------------
One way is to find the equation, then find any number of points.
Use y-y1 = m*(x-x1) where m is the slope and (x1,y1) is the point (-1,3)
y-3 = 2*(x+1) = 2x+2
y = 2x + 5
Then insert a value for x and find y
eg, x=0, y=5 --> (0,5)
x= 1/3, y = 17/3 --> (1/3,17/3)
------------
Another way is to add/subtract 2 to y for each 1 to x.
Points = (-1+2q,3+q) for any value of q

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find the x-intercept of the line with the equation 4x+2y=12.
--------------
Let y = 0, then 4x = 12 and x = 3
----
x-intercept: (3,0)
=========================
Cheers,
Stan H.

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to find the x intercept set y=0
:
4x+2(0)=12
:
4x=12
:

:
x intercept is at (3,0)

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The point-slope form of the equation can be found by using the equation
(y-y1)=m*(x-x1)
where m is the slope and {x1,y1) is the given point. Substitute the values for the corresponding variables in the general equation
y-(-2)=3/5*(x-10)
This can be simplified to y+2=3/5*(x-10)
In slope y-intercept form the equation is
y=3/5*x-3/5*10-2= 3/5x-6-2= 3/5x-8
y=(3/5)*x-8

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find the slope = 0-5/0-1=5......so m=slope=5
:
now use the point slope formula to arrive at equation....plug in any point you desire using y-k=m(x-h) where (h,k) is any point on the line....I will use (0,0)since that is the easiest
:
y-0=5(x-0)
:

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find the slope of a line with the points(1,3) and (5,4)?
---
slope = [4-3]/[5-1] = 1/4
===========================
Cheers,
Stan H.

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you use the formula In the point (1,3), 1 is your x1 and 3 is your y1. In the point (5,4), 5 is your x2 and 4 is your y2. You plug in the numbers into the formula. You solve ans simply to get your slope.

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its simply a definition.....there really is nothing to understand
:
the abcissa is the x coordinate and the ordinate is the y coordinate so in the ordered pair (3,4)
:
3 is the abcissa and 4 is the ordinate
:
in (12,-7)
:
abcissa is 12
ordinate is -7

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4x+2y=12
First, we need to put this equation into slope-intercept form which is:
y=mx+b where m is the slope and b is the y-intercept(value of y when x=0)
Subtract 4x from each side:
4x-4x+2y=-4x+12 collect like terms:
2y=-4x+12 divide each term by 2
y=-2x+6 so the slope is -2
Does this help?---ptaylor

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find the slope of the line that has the equation 4x+2y=12
2x + y = 6
y = -2x + 6
------------
slope = -2
=====================
Cheers,
Stan H.

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combine like terms and set them equal to zero
:
after subtracting 3y and 5 from both sides

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rewrite the equation 4x+4y=20 in slope-intercept form
x + y = 5
y = -x + 5
================
Cheers,
Stan H.

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A. Does the following equation have one solution, two solutions or no solutions? Explain.
(absolute value sign) x=3 (absolute value sign) +14=10

You made a typo, the = is either supposed to be a + or a -. I can't
tell which.  But it doesn't matter in this one particular case:



add  to both sides to isolate the absolute value term:



This can have no solution simply by inspection,
because no absolute value can ever be negative, so
the left side, which is an absolute value, could
never equal to the right side, because it is
negative. 

B. Solve by showing all work and then choose the correct answer

Isolate the absolute value term by adding 6 to both sides:



Remove the parentheses inside the absolute value bars:



Rule for removing absolute value bars in an equation:

 becomes two equations:
, , when N is non-negative:
(There is no solution when  is negative!)

So,  becomes:

 and 



So there are two solutions, -8 and 12, answer (a).

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

C. Solve by showing all work and then choose the correct answer

  

Divide both sides by -2 to isolate the absolute value.
Notice that when we divide both sides by a negative
numbers, we must REVERSE the inequality symbol!



The rule for removing absolute value bars when the inequality is <:
 where N is positive becomes


So


becomes

 

Add 3 to all three sides to get x in the middle:



, choice (d).

Edwin

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Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is 2 and the run is 1. This means that to go from point to point, we can go up 2 and over 1



So starting at , go up 2 units


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and

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Compare 
 

with



We see that the  is , so that makes
the y-intercept  or 
and that the slope, , is .

First we plot the y-intercept :



Now the slope is , so we
take the numerator of the slope , which
is , as the rise, and the denominator,
which is , as the run.

So we start at the y intercept and draw the rise
of  units upward, indicated below as an upward
pointing arrow.  (If the slope had been negative we 
would have drawn it pointing downward):



Then we go right  units for the run,
(run is always to the right) indicated by the second arrow
pointing to the right:



Now to graph the line, all we do is draw a
straight line through the y-intercept (0,1)
and through the arrow head at the end of
the run.  So we have this graph:



Edwin

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ok, good luck.

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Use the point-slope form of a line to write the equation.
The slope m = (60-32)/(10-0) = 28/20 = 2.8
The point-slope formula is y-32 = 2.8*(x-0) = 2.8*x
y= 2.8*x + 32
Using the graph, the predicted year in which, 100% of the workers will receive their paychecks via direct deposit is 26.
Using the equation, y = 2.8*x + 32 to make the prediction, solve the equation
100 = 2.8*x + 32 for x
100-32 = 2.8*x
68 = 2.8*x
x = 68/2.8
x = 24.3 years from 1994 = 2018.3

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Evaluate the expression y=x^3 for each given value of x
x y
-9 -729
-1 -1
4 64
5 125
Graph of the expression

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The solution to a system of two linear equations can be found by graphing each equation and determining the point of intersection of the two lines. The difficulty with using the graphical method for solving a system of two linear equations is that the solution (point of intersection) can only be estimated. The process of estimating is limited by the precision of the grid lines on the graph paper or computer screen. Note the difference in your ability to estimate the point of intersection between the two graphs.

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y=-2x+5
y=3x-10
looks as if they cross at (3,-1)
and indeed if you plug that into the equations it is a solution for both
:

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2y=-3x+8--->y=(-3/2)x+4
4y=-6x+16--->y=(-3/2)x+4
:
these are the same exact line which means there are infinite solutions
:

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First, one doesn't solve for a table of values. One determines the table of value. Since the goal is to plot the graph of an equation, one determines the table of values by computing the value of y for each value of x. Lets take a simple equation y = 5*x + 3.
When x = 0, y = 5.
When x = 1, y = 8.
When x = -1, y = -2
When x = 2, y = 13
When x = -2, y = -7.
Plot these ordered pair of points.