Questions on Algebra: Linear Equations, Graphs, Slope answered by real tutors!

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the lines y = x and y = -x + 5 are perpendicular to each other since their slopes are negative reciprocals of each other.
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the line y = 0 means that one of the sides of the triangle will be on the x-axis.
that line forms the bottom of the triangle.
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the intersection of the lines y = x, and y = -x + 5, will form the top of the triangle.
that point of intersection is found by solving y = x, and y = -x + 5 simultaneously.
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since y = x, you can substitute in the second equation to get x = - x + 5 which solves to be x = 5/2.
since x = 5/2, then y also equals 5/2, so the intersection point is (5/2,5/2)
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since the line y = x intersects the line y = 0 at the point (0,0), then (0,0) is also a point on the triangle.
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since the line y = -x + 5 intersects the line y = 0 at the point (5,0), then (5,0) is the third point of the triangle.
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if you label the triangle ABC starting from (0,0) and going around clockwise, then the 3 points on the triangle are:
A (0,0)
B (5/2,5/2)
C (5,0)
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angle B is the right angle because it is the intersects of the two perpendicular lines AB and BC.
recall that AB is on the line with the equation y = x, and BC is on the line with the equation y = -x+4, and that these lines are perpendicular to each other because the slopes of their equations are negative reciprocals of each other.
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to find the area, you need to use the formula A = 1/2 base * height.
since the right angle is at B, the base can be either AB or BC. we'll use AB.
since the base is AB, then the height is BC.
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all we need to do is find the lengths of AB and BC and we can then plug into the equation to get the area.
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the length of the line segment AB is square root of ((5/2 - 0)^2 + (5/2-0)^2).
this becomes square root of ((5/2)^2 + (5/2)^2).
this equals square root of (25/4 + 25/4) equals square root of (50/4) = 3.535533906....
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the length of the line segment BC is square root of ((5/2-0)^2 + (5/2-5)^2)
this becomes square root of ((5/2)^2 + (-5/2)^2).
this equals square root of (25/4 + 25/4) equals square root of (50/4) = 3.535533906....
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area of the triangle is 1/2 bh which equals 1/2 of .....
square root of (50/4) * square root of (50/4) which equals 1/2 of .....
(50/4) which equals ...
50/8 which equals 6.25.
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area of the triangle is 50/8 = 6.25.
---
a graph of the triangle is shown below.
look below the graph for further comments.

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(5/2,5/2) is where y=x and y=-x+5 meet
:
we know that the hypothenuse is 5 units
we also know that the 2 legs are equal. lets call each leg x
:
so
:

:

:

:

:
A=1/2bh
:
we know the base and height are the length of the legs
:

:






:

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Start with the given equation.


Add to both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.





In order to graph this equation, we only need two points to create a straight line




--------------------------------Let's find the first point--------------------------------

Start with the given equation


Plug in


Multiply and to get


Add


Reduce



So when , we have the value . This means we have the first point




--------------------------------Let's find the second point--------------------------------

Start with the given equation


Plug in


Multiply and to get


Add


Reduce



So when , we have the value . This means we have the second point




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


So we have the two points: and


Now plot these two points on a coordinate system





Now draw a straight line through the two points. This line is the graph of

Graph of through the two points and

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We can see that the equation is in slope intercept form where the slope is and the y-intercept is note: the y-intercept is the point



Since we're looking at the point (-4,-10), this means that and


Start with the given equation.


Plug in and


Multiply


Multiply 12 by


Subtract the fractions.


Since the equation is NOT true, this means that the point (-4,-10) is NOT on the line . So the flamingo at the point (-4,-10) is NOT on the water line


So you can place the flamingo at the point (-4,-10).

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A) the slope has to be the same to be parallel m the slope for y=x+3 is 1. Now we have the slope and a point. using the point slope formula
:
y-0=1(x-5)
:

:
B)in y=2x+3 the slope is 2. using the point slope formula
:
y-1=2(x+4)
:
y-1=2x+8
:

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Replace x with -4 and y with -10 in the given equation and then simplify.
If you get the same answer on both sides of the equation, then the answer is yes. If not, then answer is no.
Understand?
=================================
I received your e-mail. Yes, I also got no. Like I said, they give you a point. Each point has a value for x and y as you know. The question would like to know if the given point makes the equation true.
After replacing x and y with their value and simplifying the equation, we learn that the given point (-4, -10) does not get the same answer on both sides of the equation.
Understand?

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m=slope=rise/run= well you get the point
:
m=3-0/0-3=-1
:
y intercept 3
:
y=-x+3

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need a question!!! thanks

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you have two points and from 2 points you can find the slope m
:
4-0/0-(-1)=4
:
y intercept is 4
:

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See answer to Problem 175521. Do this one the same way.

http://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.175521.html

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m=slope =-8-0/0-(-4)=-8/-4=-2..... y intercept is -8
:
so

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See answer to Problem 175521. Do this one the same way.

http://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.175521.html

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The x-intercept is the point where the y coordinate is zero. Likewise, the y-intercept is the point where the x coordinate is zero. Therefore the two points described are: and

Now use the two-point form of the equation of a line:



Just substitute and do the arithmetic:



I'll let you do the arithmetic part.

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b)



If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation:


---Point-Slope Formula---
where is the slope, and is the given point

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)


Distribute

Multiply and to get

Add 3 to both sides to isolate y

Combine like terms and to get
------------------------------------------------------------------------------------------------------------
Answer:


So the equation of the line with a slope of which goes through the point (,) is:

which is now in form where the slope is and the y-intercept is

Notice if we graph the equation and plot the point (,), we get (note: if you need help with graphing, check out this solver)

Graph of through the point (,)
and we can see that the point lies on the line. Since we know the equation has a slope of and goes through the point (,), this verifies our answer.

---

c)




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


Distribute


Multiply


Add 5 to both sides.


Combine like terms. note: If you need help with fractions, check out this solver.


Simplify


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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x-intercept:2 tells us that the point (2,0) is on the line
y-intercept: -4 means that (0,-4) is on the line


So we need to find the equation of the line through the points (2,0) and (0,-4)





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


Distribute


Multiply


Add 0 to both sides.


Combine like terms.


Simplify


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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If you want to find the equation of line with a given a slope of which goes through the point (-5,4), you can simply use the point-slope formula to find the equation:


---Point-Slope Formula---
where is the slope, and is the given point

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)


Rewrite as


Distribute


Multiply and to get


Multiply and to get


Add 4 to both sides to isolate y


Combine like terms.
------------------------------------------------------------------------------------------------------------
Answer:


So the equation of the line with a slope of which goes through the point (,) is:

which is now in form where the slope is and the y-intercept is

Notice if we graph the equation and plot the point (-5,4), we get


Graph of through the point (-5,4)
and we can see that the point lies on the line. Since we know the equation has a slope of and goes through the point (-5,4), this verifies our answer.

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If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation:


---Point-Slope Formula---
where is the slope, and is the given point

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)


Rewrite as


Rewrite as


Distribute

Multiply and to get

Subtract 3 from both sides to isolate y

Combine like terms and to get
------------------------------------------------------------------------------------------------------------
Answer:


So the equation of the line with a slope of which goes through the point (,) is:

which is now in form where the slope is and the y-intercept is

Notice if we graph the equation and plot the point (,), we get (note: if you need help with graphing, check out this solver)

Graph of through the point (,)
and we can see that the point lies on the line. Since we know the equation has a slope of and goes through the point (,), this verifies our answer.

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If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation:


---Point-Slope Formula---
where is the slope, and is the given point

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)


Rewrite as


Distribute

Multiply and to get

Subtract 4 from both sides to isolate y

Combine like terms and to get
------------------------------------------------------------------------------------------------------------
Answer:


So the equation of the line with a slope of which goes through the point (,) is:

which is now in form where the slope is and the y-intercept is

Notice if we graph the equation and plot the point (,), we get (note: if you need help with graphing, check out this solver)

Graph of through the point (,)
and we can see that the point lies on the line. Since we know the equation has a slope of and goes through the point (,), this verifies our answer.

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If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation:


---Point-Slope Formula---
where is the slope, and is the given point

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)


Distribute

Multiply and to get

Add 2 to both sides to isolate y

Combine like terms and to get
------------------------------------------------------------------------------------------------------------
Answer:


So the equation of the line with a slope of which goes through the point (,) is:

which is now in form where the slope is and the y-intercept is

Notice if we graph the equation and plot the point (,), we get (note: if you need help with graphing, check out this solver)

Graph of through the point (,)
and we can see that the point lies on the line. Since we know the equation has a slope of and goes through the point (,), this verifies our answer.

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X-intercept:1 means that the point (1,0) is on the line

Y-intercept:2 means that the point (0,2) is on the line


So let's find the equation of the line through the points (1,0) and (0,2)


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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this is a system of equation that we will solve similtaneously
:
-3a + b = 4 ....eq 1
-9a + 5b = -1...eq 2
:
multiply eq 1 by -3
:
9a-3b=-12....revised eq 1
-9a+5b=-1....eq 2
:
I lined the 2 equations up so you can see that when we add the equations together the a terms are eliminated because 9a-9a=0. We are left with
-3b+5b=-12-1.
:
2b=-13
:

:
now take b's found value and plug it back into either equation. I will use eq 1
:
-3a + (-13/2)=4 ........now multiply all terms by 2 to get rid of fraction
:
-6a-13=8
:
-6a=21
:

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Start with the given system of equations:



Multiply the both sides of the first equation by -3.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms. Notice how the x terms cancel out.


Simplify.


Divide both sides by to isolate .


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


Now go back to the first equation.


Plug in .


Multiply.


Multiply both sides by the LCD to clear any fractions.


Distribute and multiply.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So our answer is and .


Which form the ordered pair .


This means that the system is consistent and independent.

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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


Rewrite as


Distribute


Multiply


Subtract 3 from both sides.


Combine like terms. note: If you need help with fractions, check out this solver.


Simplify


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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The average lifespan of American women has been tracked, and the model for the data is y = 0.2t + 73, where t = 0 corresponds to 1960.
What is the slope? It is m = 0.2. This values tells me that, for every increase of 1 in my input variable t (that is, for every increase of one year), the value of my output variable y will increase by 0.2.

What is the meaning of the slope? It means that, every year, the average lifespan of American women increased by 0.2 years, or about 2.4 months.
When t = 0, what is the value of y? Looking at the equation, I see that y = 73.
What is the meaning of this y-value? It means that, in 1960 (when they started counting), the average lifespan of an American woman was 73 years.

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Start with the first equation.


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.


So we can see that the equation has a slope and a y-intercept .


Now move onto the second equation.


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.


So we can see that the equation has a slope and a y-intercept .


So the slope of the first line is and the slope of the second line is .


Notice how the slope of the second line is simply the negative reciprocal of the slope of the first line .


In other words, if you flip the fraction of the second slope and change its sign, you'll get the first slope. So this means that and are perpendicular lines.

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Rewrite both equations into slope intercept form.
1.x+4=y
y-x=-3
y=x+4 (line #1)
y=x-3 )line #2)
What is the slope of each line?
The slope of line 1 is 1
The slope of line 2 is 1.
Yes, the line are parallel.
Here is a graph of the two lines.

The lines differ because of different y-intercepts.
I hope that this helps.

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Solve both equations for , in other words, perform whatever algebraic manipulations you need to have on one side of the equal sign and everything else on the other side. Once you have done that, if the coefficient on the in both equations is exactly the same, then the lines are parallel -- otherwise not.

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select any two points on the line
(0,2)and(1,4)
------------------
slope = (4-2)/(1-0) = 2
intercept: (0,2)
----------
Equation:
y = 2x + 2
-----------
Points: (1,4); (3,8)
==========================
Cheers,
Stan H.

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answer

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Find the slope of the line at passes through (3,6) and (1,-4).
----------------------
slope = (6--4)/(3-1) = 10/2 = 5
==================================
Cheers,
Stan H.

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p(n) = 45n-2250
===================
Cheers,
Stan H.

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Find the slope of the line that passes through the points (-3, -5) and (-5, 1).
--------------
The slope, m, is the "rise over the run", the (diff in y)/(diff in x)
m = (-5-1)/(-3 - (-5))
m = -6/2
m = -3

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if the equation is 2y=x+4 and y=0 then 2(0)=x+4
:
0=x+4
:
x=-4
:
therefore h=-4 since h=x and we have the ordered pair(-4,0)

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.
(h,1)------------>this is in point form (x,y)

plug h in for x and 1 in for y:



Now simplify



Now subtract 4 from each side

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Let E be the Domain{20,30,45,60} then the range is L
:
L=120-e
:
20: L=120-20=100
30: L=120-30=90
45: L=120-45=75
60: L=120-60=60
:
ordered pairs (20,100),(30,90),(45,75),(60,60)
:
If L is the Domain these ordered pairs would be reversed!!

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if the equation is 2y=x+4 and y=2 then 2(2)=x+4
:
4=x+4
:
x=0
:
therefore h=0 since h=x and we have the ordered pair(0,2)

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Find given that

Substitute 9 for all the x's"



Simplify the right side:





Easy, wasn't it?

Edwin

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Find given that

Erase all the 's



Put 's where the 's were:



Simplify the right side:





Easy, wasn't it?

Edwin

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Assume that y varies directly as x.
y = kx
---------
Find y when x=54
if y=1/4 when x=3/2.
(1/4) = k(3/2)
k = (2/3)(1/4) = 1/6
---
y = (1/6)54
y = 9
============
Cheers,
Stan H.

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Assume that y varies inversely as x:
y = k/x
--------------------
y=18 when x=3
18 = k/3
--------------------
Find the constant of variation.
k = 3*18 = 54
===================
Cheers,
Stan H.

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Yes, if all the powers that the unknowns are raised to
are , which they are:

then it is linear
The standard form is

Subtract from both sides

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A verticle line has an undefined slope.
x=5 is the equation for this line.

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To be at the intersection, the given point would have to be
on both lines

It's on this line


not true, this point lies outside the line