# SAT Mathematics : Graphing Linear Equations

## Example Questions

### Example Question #1 : Graphing Linear Equations

Line A is represented by the line . Which of the following represents the equation of a line that is perpendicular to Line A and that travels through point ?

Explanation:

This problem tests two key principles related to lines in the coordinate plane. For one, the slopes of perpendicular lines are negative reciprocals of each other (e.g.  and . So here since you need to find a line perpendicular to a line with slope of , you're looking for the negative reciprocal . You can therefore eliminate choices  and , which do not have the proper slope.

The second key concept relates to what a point in the coordinate plane is. Because points are all given in the form  you know that for a point to pass through the point  it will need to allow for  to equal  when  equals . To find the  term (in ), take what you do know about the line:

, so the correct equation is .

### Example Question #31 : Graphical Representation Of Functions

Which of the following equations represents a line that is perpendicular to ?

Explanation:

In order for two lines to be perpendicular, the slopes need to be negative reciprocals of each other. Since the given slope is , you're looking for a slope of .

To find the slope, you need to put the equations in point-slope form, , where  is the slope. Checking the answer choices quickly:

Choices  and  will each give  a positive coefficient, as your first step is to get  on the opposite side of the equation from .

Only choice  simplifies to a coefficient of :

### Example Question #2 : Graphing Linear Equations

Line J has a y-intercept of 6 and passes through point . What is the slope of Line J in terms of  and ?

Explanation:

You know that the line goes through point  and the y-intercept of 6 tells you that the line also goes through point . So you can use the "rise over run" slope formula to calculate the slope. Recognizing that the five answer choices focus on , and 6, you should see that you should subtract 0 and 6 (subtracting 0 is the same as doing nothing at all) to get your math looking like the answer choices. So you'll set up the equation:

### Example Question #4 : Graphing Linear Equations

In the coordinate plane, Line J passes through the origin as well as points  and . Which of the following could be the slope of Line J?

Explanation:

That Line J passes through the origin gives you a very helpful bit of information regarding point-slope form . That tells you that when , meaning that  meaning that  also equals . That takes a variable away, allowing you to get to work on the two points knowing that the line has a constant slope, :

For point , that means that .

For point , that means that .

This means that you can plug in  into the first equation, allowing you to solve for  (the slope):

(or ).

### Example Question #5 : Graphing Linear Equations

In the coordinate plane, Line A has a slope of -1 and an x-intercept of 1. Line B has a slope of 2 and a y-intercept of -2. If the two lines intersect at the point , what is the sum ?

Explanation:

When attempting to find where two lines intersect, it is typically best to get the lines in the form . For Line A, you know that the slope is -1, so you have a head start in that . So you're starting with . And then remember - the x-intercept is the point at which , so that point is . You can then plug that point into the equation to find :

, so . You know now that Line A has the equation .

For Line B, you know that the slope is 2 (so ) and that when  is , . Plug that into the line equation to solve for b and you have -2 = 2(0) + b, so b = -2. Now you know the equation for Line B: .

Since  and , the two lines will intersect where . Algebraically that leads you to , so . Plug that back into either line to find y, and you'll find that . Since the point of intersection is , the sum  makes  the correct answer.

### Example Question #6 : Graphing Linear Equations

The graph of a linear equation is shown in the xy-plane above. The slope of the graph of the linear equation , where  and  are constants, is twice the slope of the graph of the given equation. If the graph of  passes through the point , what is the y-intercept of the new line ?

Explanation:

We’ll need to use the graph of the linear equation to better understand the existing line, so that we can draw conclusions about the new line. Since the existing line moves vertically by one for every three it moves to the right, our slope, or  is Since our line crosses the y axis at the point (0, 4), our full equation is .

Thus, our new line will have a slope of  since the question told us that the slope of our new line, , will be twice the slope of the pictured line. Now, to find the full equation of the new line, and thus its y-intercept, we’ll want to plug the known point of the line  into the equation.

If , we can solve for !

, thus .

So, our new line is  (shown below in blue) and its y-intercept is .

### Example Question #7 : Graphing Linear Equations

The line  , where m is a constant, is graphed in the xy-plane. If the line contains the point , where  and , what is the slope of the line in terms of  and ?

Explanation:

To identify the slope of the line, we’ll want to identify two points on the line to find the  otherwise known as the One point,  is given to us. Our other point can be determined by looking to the y-intercept, in this case, . So, if we identify our  by taking , we arrive at  or , our correct answer.

### Example Question #8 : Graphing Linear Equations

Which of the following points does the line  NOT pass through?

(-5, -3.5)

(14, 25)

(2, 7)

(6, 12)

(6, 12)

Explanation:

In order to attack this question, we can plug in the value of x for each coordinate to see if the proper value of y matches. for

(2, 7) if we plug in 2 for x, we can see that

which does in fact = 7.

For (-5, -3.5), if we take the same step we can see that

does give us a y value of -3.5.

For (14, 25), plugging in x = 14 gets us to

which simplifies to 25.

So, by default we can see that our correct answer must be

(6, 12), since an input of 6 gives us

which equals 13, not 12. So, (6, 12) is the only point *not* on the line.

### Example Question #1 : Graphing Linear Equations

What is the area of the triangle formed by the lines , and ?

15

48

6

45

45

Explanation:

In order to find the area of the triangle, we can use relevant points such as the point of intersection and the x intercepts. In this case, the point of intersection is the point at which the equations of the lines are equal. So, if

then x = 4, and if we plug 4 back into our equation we can see that y = 6. So, the point of intersection is (4, 6).

If we set y = 0 for each equation, we can also find the x intercepts, in this case, (1, 0) and (16, 0).

The triangle will appear as follows

At this point, we can use the x intercepts to find the base, 16-1 or 15, and the point of intersection to find the height, in this case 6.

If the area of a triangle is

we can fill in the unknowns with

to arrive at an area of 45.

### Example Question #10 : Graphing Linear Equations

The equation 3x+2y=6 represents a line. This line does NOT pass through which of the four quadrants?

IV

II

I

III

III

Explanation:

We can quickly visualize this line to draw conclusions.

In order to do so, plug in 0 for x to find a point on the line:

3(0)+2y=6

y=3

Thus, (0,3) is a point on the line.

Plug in 0  for y to find a second point on the line:

3x+2(0)=6

x=2

(2,0) is another point on the line.

Now we know that the line passes through the points (2,0) and (0,3).

A quick sketch of the two points reveals that the line passes through all but the third quadrant.

*note - a line with a positive slope will always pass through quadrants I and III, while a line with a negative slope will always pass through quadrants II and IV.*