A solution to the system of equations is . Which of the following is a possible value of ?

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SAT Nonlinear Equations & Systems Question 207 | Free SAT Prep | Aniko

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Question 207·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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x^2 - y = 4

y = 2x - 4

A solution to the system of equations is $(x,y)$ . Which of the following is a possible value of $x - y$ ?

For nonlinear systems where one equation is already solved for a variable (like $y = 2x - 4$ ), quickly substitute that expression into the other equation to eliminate a variable. Solve the resulting quadratic carefully, remembering that factoring can produce two solutions. Then plug each $x$ back into the simpler equation to find $y$ , compute the specific expression the question asks for (here $x - y$ ) for each solution pair, and only then compare these values to the answer choices so you don’t accidentally stop at just one solution or mix up $x - y$ with another expression.

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Eliminate one variable

You already have $y$ written in terms of $x$ as $y = 2x - 4$ . Try substituting this expression for $y$ into the other equation to form an equation with only $x$ .

Solve carefully for x

After you substitute, simplify the equation. You should get a quadratic in $x$ that can be factored. Remember that a factored quadratic like $x(a - b) = 0$ can have two solutions.

Use both solutions and then form x − y

For each $x$ you find, plug it into $y = 2x - 4$ to get $y$ . Then compute $x - y$ for each pair and see which result is listed in the answer choices.

Desmos Guide

Graph both equations

Rewrite the first equation as $y = x^2 - 4$ . In Desmos, enter y = x^2 - 4 and y = 2x - 4 as two separate functions so you can see where they intersect.

Find the intersection points

Click on each point where the two graphs intersect. Desmos will display the coordinates $(x,y)$ of each intersection; these are the solutions to the system.

Compute x − y for each intersection

For each intersection point you see, take its $x$ - and $y$ -coordinates and in a new expression box type x_value - y_value using those numbers (for example, if one point were $(a,b)$ you would type a - b). Compare the resulting values with the answer choices and identify which choice matches one of these $x - y$ values.

Step-by-step Explanation

Use substitution to get one equation in one variable

We already know $y$ in terms of $x$ : $y = 2x - 4$ .

Substitute this into the first equation $x^2 - y = 4$ :

x^2 - (2x - 4) = 4

Now simplify this equation.

Solve the resulting quadratic for x

Simplify the equation:

x^2 - 2x + 4 = 4

Subtract 4 from both sides:

x^2 - 2x = 0

Factor:

x(x - 2) = 0

So there are two possible $x$ -values that satisfy the system.

Find the corresponding y-values for each x

From the factored equation we have two cases:

If $x = 0$ , then use $y = 2x - 4$ to find $y$ .
If $x = 2$ , then use $y = 2x - 4$ to find $y$ .

This gives you two solution pairs $(x,y)$ to the system.

Compute x − y for each solution and compare to the choices

For each solution pair $(x,y)$ that you found in Step 3, calculate $x - y$ .

You will get two different values of $x - y$ ; only one of them appears in the answer choices. The value that matches a choice is $\boxed{2}$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation