A system of equations is given by If is a solution to the system, what is one possible value of ?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 92 | Free SAT Prep | Aniko

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Question 92·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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A system of equations is given by

\begin{aligned} x + y &= 10 \\ x^2 + y &= 40 \end{aligned}

If $(x,y)$ is a solution to the system, what is one possible value of $y$ ?

For systems with one linear and one nonlinear equation, use substitution: solve the linear equation for one variable, substitute into the nonlinear equation to get a single-variable equation, then solve. Factor when possible (look for integer factor pairs), or use the quadratic formula if needed. Finally, substitute back to find the other variable.

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Use the simpler equation first

The first equation $x + y = 10$ is linear. Can you rewrite it to express $y$ in terms of $x$ ?

Substitute into the second equation

Once you have $y$ in terms of $x$ , replace $y$ in $x^2 + y = 40$ with that expression. What type of equation do you get?

Factor the quadratic

After substitution, you should have $x^2 - x - 30 = 0$ . Look for two numbers that multiply to $-30$ and add to $-1$ .

Find y from x

Once you find the $x$ values, plug each back into $y = 10 - x$ to find the corresponding $y$ values.

Desmos Guide

Graph both equations

Enter the equations in function form:

y = 10 - x
y = 40 - x^2

Find the intersection points

Look for the points where the line and parabola intersect. Click on each intersection point to see its coordinates.

Read the y-coordinates

The $y$ -coordinates of the intersection points are the possible values of $y$ . One of these values should match an answer choice.

Step-by-step Explanation

Solve the linear equation for one variable

From $x + y = 10$ , solve for $y$ :

y = 10 - x.

Substitute into the second equation

Replace $y$ in $x^2 + y = 40$ :

x^2 + (10 - x) = 40.

Simplify:

x^2 - x - 30 = 0.

Factor and solve for x

Factor the quadratic:

(x - 6)(x + 5) = 0.

So $x = 6$ or $x = -5$ .

Find the corresponding y-values

Using $y = 10 - x$ :

For $x = 6$ : $y = 10 - 6 = 4$
For $x = -5$ : $y = 10 - (-5) = 15$

The answer choice $4$ is correct.

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Step-by-step Explanation

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