SAT Math : Geometry
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Example Questions
Example Question #1 : X And Y Intercept
Solve the equation for x and y.
x² + y = 31
x + y = 11
x = 13, 7
y = 8, –6
x = 6, 15
y = 5, –4
x = 5, –4
y = 6, 15
x = 8, –6
y = 13, 7
x = 5, –4
y = 6, 15
Solving the equation follows the same system as the first problem. However since x is squared in this problem we will have two possible solutions for each unknown. Again substitute y=11-x and solve from there. Hence, x2+11-x=31. So x2-x=20. 5 squared is 25, minus 5 is 20. Now we know 5 is one of our solutions. Then we must solve for the second solution which is -4. -4 squared is 16 and 16 –(-4) is 20. The last step is to solve for y for the two possible solutions of x. We get 15 and 6. The graph below illustrates to solutions.
Example Question #581 : Sat Mathematics
Solve the equation for x and y.
x² – y = 96
x + y = 14
x = 15, 8
y = 5, –14
x = 5, –14
y = 15, 8
x = 25, 4
y = 10, –11
x = 10, –11
y = 25, 4
x = 10, –11
y = 25, 4
This problem is very similar to number 2. Derive y=14-x and solve from there. The graph below illustrates the solution.
Example Question #2 : How To Find The Slope Of Perpendicular Lines
Solve the equation for x and y.
5x² + y = 20
x² + 2y = 10
x = √10/3, –√10/3
y = 10/3
No solution
x = √4/5, 7
y = √3/10, 4
x = 14, 5
y = 4, 6
x = √10/3, –√10/3
y = 10/3
The problem involves the same method used for the rest of the practice set. However since the x is squared we will have multiple solutions. Solve this one in the same way as number 2. However be careful to notice that the y value is the same for both x values. The graph below illustrates the solution.
Example Question #3 : How To Find The Slope Of Perpendicular Lines
Solve the equation for x and y.
x² + y = 60
x – y = 50
x = 10, –11
y = –40, –61
x = 11, –10
y = 40, 61
x = 40, 61
y = 11, –10
x = –40, –61
y = 10, –11
x = 10, –11
y = –40, –61
This is a system of equations problem with an x squared, to be solved just like the rest of the problem set. Two solutions are required due to the x2. The graph below illustrates those solutions.
Example Question #1 : X And Y Intercept
A line passes through the points 
None of the available answers
First we will calculate the slope as follows:
And our equation for a line is
Now we need to calculate b. We can pick either of the points given and solve for
Our equation for the line becomes
Example Question #1 : How To Find X Or Y Intercept
If the equation of a line is 4y – x = 48, at what point does that line cross the x-axis?
(0,–48)
(0,–12)
(0,12)
(–48,0)
(48,0)
(–48,0)
When the equation crosses the x-axis, y = 0. Plug 0 into the equation for y, and solve for x.
4(0) – x = 48, –x = 48, x = –48
Example Question #2 : How To Find X Or Y Intercept
The slope of a line is equal to -3/4. If that line intersects the y-axis at (0,15), at what point does it intersect the x-axis?
15
5
60
20
-20
20
If the slope of the line m=-3/4, when y=15 and x=0, plug everything into the equation y=mx+b.
Solving for b:
15=(-3/4)*0 + b
b=15
y=-3/4x + 15
To get the x-axis intersect, plug in y=0 and solve for x.
0 = -3/4x + 15
3/4x = 15
3x = 15*4
x = 60/3 = 20
x=20
Example Question #1 : How To Find X Or Y Intercept
If these three points are on a single line, what is the formula for the line?
(3,3)
(4,7)
(5,11)
y = 4x + 31
y = 3x - 3
y = 5x + 11
y = 3x - 9
y = 4x - 9
y = 4x - 9
Formula for a line: y = mx + b
First find slope from two of the points: (3,3) and (4,7)
m = slope = (y2 – y1) / x2 – x1) = (7-3) / (4-3) = 4 / 1 = 4
Solve for b by plugging m and one set of coordinates into the formula for a line:
y = mx + b
11 = 4 * 5 + b
11 = 20 + b
b = -9
y = 4x - 9
Example Question #1 : X And Y Intercept
The slope of a line is 5/8 and the x-intercept is 16. Which of these points is on the line?
(8,15)
(16, 10)
(0,10)
(32,30)
(32,10)
(32,10)
y = mx + b
x intercept is 16 therefore one coordinate is (16,0)
0 = 5/8 * 16 + b
0 = 10 + b
b = -10
y = 5/8 x – 10
if x = 32
y = 5/8 * 32 – 10 = 20 – 10 = 10
Therefore (32,10)
Example Question #4 : How To Find X Or Y Intercept
A line has the equation: x+y=1.
What is the y-intercept?
0.5
0
1
-1
2
1
x+y=1 can be rearranged into: y=-x+1. Using the point-slope form, we can see that the y-intercept is 1.
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