How to find the equation of a curve - SAT Math

Card 0 of 5

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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