Consider the system of equations What is the positive value of that satisfies the system?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 77 | Free SAT Prep | Aniko

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Question 77·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Consider the system of equations

\begin{cases} x + y = 5 \\ x^{2} - y = 7 \end{cases}

What is the positive value of $x$ that satisfies the system?

For systems where one equation is linear and the other is nonlinear (like a quadratic), use substitution: solve the linear equation for one variable, substitute into the nonlinear equation to get a single-variable equation, then solve it (usually by factoring or the quadratic formula). Finally, apply any extra conditions from the question (such as “positive,” “integer,” or “greater than”) to pick the correct solution from the ones you found, and quickly check that it satisfies both original equations.

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Relate the two equations

Try to solve one of the equations for $y$ and substitute that expression into the other equation so you only have one variable.

Form a single-variable equation

From $x + y = 5$ , write $y$ in terms of $x$ and plug that into $x^2 - y = 7$ . Simplify carefully.

Solve the resulting quadratic

You should get a quadratic equation in $x$ . Factor it (or use the quadratic formula) to find both solutions, then think about which one the question is asking for.

Desmos Guide

Graph both equations as functions of $x$ and $y$

In Desmos, enter the first equation as y = 5 - x and the second equation as y = x^2 - 7. These represent the same equations as in the system.

Find the intersection points

Look for the intersection points of the line and the parabola. Click on each intersection; Desmos will display the coordinates $(x, y)$ of those points.

Identify the required value of $x$

Among the $x$ -coordinates of the intersection points, note which ones appear. The problem asks for the positive $x$ -value from these intersections; that is the value you should choose from the answer options.

Step-by-step Explanation

Use the first equation to express $y$ in terms of $x$

From the first equation,

x + y = 5

solve for $y$ :

y = 5 - x.

Substitute into the second equation to get one equation in $x$

The second equation is

x^2 - y = 7.

Substitute $y = 5 - x$ :

x^2 - (5 - x) = 7.

Simplify the left side:

x^2 - 5 + x = 7.

Bring all terms to one side:

x^2 + x - 12 = 0.

Factor the quadratic equation

Factor $x^2 + x - 12$ :

x^2 + x - 12 = (x + 4)(x - 3) = 0.

So the possible values of $x$ are:

x + 4 = 0 \text{ or } x - 3 = 0,

which gives

x = -4 \text{ or } x = 3.

Use the condition on $x$ to choose the correct solution

The problem asks for the positive value of $x$ that satisfies the system. From the two solutions $x = -4$ and $x = 3$ , the positive one is $x = 3$ .

So the correct answer is $3$ .

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