Consider the system of equations Which of the following is the positive value of that satisfies the system?

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SAT Nonlinear Equations & Systems Question 86 | Free SAT Prep | Aniko

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Question 86·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Consider the system of equations

\begin{cases} x^2 + y = 11 \\ y - x = 1 \end{cases}

Which of the following is the positive value of $x$ that satisfies the system?

For systems with one linear equation and one nonlinear (like a quadratic), first solve the linear equation for one variable, usually the one that is easiest to isolate. Substitute this expression into the nonlinear equation to reduce the system to a single equation in one variable. Simplify to a standard quadratic, then solve by factoring if possible or by using the quadratic formula, and finally apply any conditions in the question (such as choosing the positive solution) to select the correct answer from the choices.

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Start with the easier equation

Look at the equation $y - x = 1$ . Can you solve this quickly for $y$ in terms of $x$ ?

Reduce the system to one variable

After you have $y$ written in terms of $x$ , substitute that expression for $y$ into the other equation so that you only have $x$ left.

Recognize and solve the quadratic

Once you substitute, you should get a quadratic equation in $x$ . If it does not factor nicely, use the quadratic formula.

Use the condition in the question

You will get two values for $x$ . Check which one is positive and select that one from the choices.

Desmos Guide

Graph both equations

In Desmos, enter the first equation as y = 11 - x^2 (solving $x^2 + y = 11$ for $y$ ) and the second equation as y = x + 1.

Find the intersection points

Look for the points where the two graphs intersect. There should be two intersection points; click each one to see its coordinates.

Identify the positive $x$ -value

Among the intersection points, note the $x$ -coordinate that is positive. That positive $x$ -value corresponds to the correct answer choice.

Step-by-step Explanation

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

From the second equation,

y - x = 1,

add $x$ to both sides to solve for $y$ :

y = x + 1.

Substitute into the first equation to get an equation in $x$ only

The first equation is

x^2 + y = 11.

Substitute $y = x + 1$ into this equation:

x^2 + (x + 1) = 11.

Combine like terms:

x^2 + x + 1 = 11.

Subtract 11 from both sides:

x^2 + x - 10 = 0.

Now you have a quadratic equation in $x$ .

Apply the quadratic formula

For the quadratic equation $x^2 + x - 10 = 0$ , the coefficients are $a = 1$ , $b = 1$ , and $c = -10$ .

Use the quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Substitute $a = 1$ , $b = 1$ , and $c = -10$ :

x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-10)}}{2(1)}.

Simplify inside the square root:

1^2 - 4(1)(-10) = 1 + 40 = 41,

x = \frac{-1 \pm \sqrt{41}}{2}.

Choose the positive solution and match it to the answer choices

There are two solutions:

x = \frac{-1 + \sqrt{41}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{41}}{2}.

Since $\sqrt{41} \approx 6.4$ , the value $\dfrac{-1 + \sqrt{41}}{2}$ is positive, and $\dfrac{-1 - \sqrt{41}}{2}$ is negative.

The question asks for the positive value of $x$ , so the correct choice is

\boxed{\dfrac{-1 + \sqrt{41}}{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