x and y Intercept - SAT Math

Card 0 of 20

Question

Solve the equation for x and y.

x² + y = 31

x + y = 11

Answer

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.

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Question

Solve the equation for x and y.

x² – y = 96

x + y = 14

Answer

This problem is very similar to number 2. Derive y=14-x and solve from there. The graph below illustrates the solution.

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Question

Solve the equation for x and y.

5_x_² + y = 20

x_² + 2_y = 10

Answer

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.

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Question

Solve the equation for x and y.

_x_² + y = 60

x – y = 50

Answer

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.

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Question

A line passes through the points $(3,5)$ and $(4,7)$ . What is the equation for the line?

Answer

First we will calculate the slope as follows:

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{7-5}{4-3}=\frac{2}{1}=2$

And our equation for a line is

$y=mx+b=2x+b$

Now we need to calculate b. We can pick either of the points given and solve for $\dpi{100} b$

$5=2(3)+b$

$b=-1$

Our equation for the line becomes

$y=2x-1$

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Question

Which of the following lines does not intersect the line ?

Answer

Parallel lines never intersect, so you are looking for a line that has the same slope as the one given. The slope of the given line is –4, and the slope of the line in y = –4_x_ + 5 is –4 as well. Since these two lines have equal slopes, they will run parallel and can never intersect.

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Question

If the equation of a line is 4_y_ – x = 48, at what point does that line cross the x-axis?

Answer

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

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Question

Where does the graph of 2x + 3y = 15 cross the x-axis?

Answer

To find the x-intercept, set y=0 and solve for x. This gives an answer of x = 7.5.

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Question

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?

Answer

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 = 154

x = 60/3 = 20

x=20

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Question

Find the y-intercept of .

Answer

To find the y-intercept, set x equal to zero and solve for y.

This gives y = 3(0)2 + 2(0) +7 = 7.

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Question

If these three points are on a single line, what is the formula for the line?

(3,3)

(4,7)

(5,11)

Answer

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

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Question

The slope of a line is 5/8 and the x-intercept is 16. Which of these points is on the line?

Answer

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)

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Question

A line has the equation: x+y=1.

What is the y-intercept?

Answer

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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Question

A line has the equation: 2x+4y=8.

What is the x-intercept?

Answer

To find the x-intercept, rearrange the equation 2x+4y=8 so that x is isolated:

2x=-4y+8

x=-2y+4

Using the point-slope formula, we see that the x-intercept is 4.

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Question

Given the line , what is the sum of the -intercept and the -intercept?

Answer

Intercepts occur when a line crosses the -axis or the -axis. When the line crosses the -axis, then and . When the line crosses the -axis, then and . The intercept points are and . So the -intercept is and the intercept is and the sum is .

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Question

What is the y intercept of the following function of x?

y = 3x

Answer

The answer is 0 because in slope intercept form, y = mx + b; b is the y intercept. In this case b = 0.

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Question

What is the x-intercept of a line with a slope of 5 and y-intercept of 3.5?

Answer

To solve this, first find the equation of our line. The form of the question gives it to us very directly. We can use the slope-intercept form (y = mx + b).

y = 5x + 3.5

The x-intercept is found by setting y = 0, because that will give us the x-value at which the line crosses the x-axis.

0 = 5x + 3.5; –3.5 = 5x; x = –3.5 / 5 or –0.7. The point will be (–0.7, 0)

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Question

Determine the y-intercept of the following line:

$\dpi{100} \small 3x+6y=9$

Answer

The y-intercept occurs when $\dpi{100} \small x=0$

$\dpi{100} \small 3x+6y=9$

$\dpi{100} \small 3(0)+6y=9$

$\dpi{100} \small 0+6y=9$

$\dpi{100} \small y = \frac{9}{6}=1.5$

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Question

At what point does the graph $3y-2x=31$ cross the -axis?

Answer

The graph crosses the -axis where $x=0$ . So plugging in and solving yields $\frac{31}{3}$ .

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Question

Find the x-intercepts of $25x^{2}+4y^{2} = 9$ .

Answer

To find the x-intercepts, plug $y=0$ into the equation and solve for $x$ .

$25x^{2} + 4\cdot 0^{2} = 9$

$25x^{2} = 9$

$x^{2} = \frac{9}{25}$

$x = \pm \frac{3}{5}$

Don't forget that there are two solutions, both negative and positive!

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