SAT Nonlinear Equations & Systems Question 227 | Free SAT Prep | Aniko

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Question 227·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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\begin{aligned} 2x + y &= 5 \\ x^2 + y^2 &= 10 \end{aligned}

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

Your answer

(Express the answer as an integer)

For a system where one equation is linear and the other is a circle (or another nonlinear equation), quickly solve the linear equation for one variable and substitute into the nonlinear equation. This reduces the problem to a single-variable quadratic, which you can usually factor or solve with the quadratic formula. Always read the question carefully to see whether it wants a specific solution (like the greater $x$ -value) rather than just any solution, and choose the appropriate root accordingly.

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Use the simpler equation first

One of the equations is linear. Try solving it for $y$ (or $x$ ) so you can substitute into the other equation.

Substitute into the other equation

After you express $y$ in terms of $x$ from the first equation, plug that expression into $x^2 + y^2 = 10$ so that the second equation only has $x$ .

Solve the resulting quadratic carefully

You should get a quadratic equation in $x$ . Factor it or use the quadratic formula to find both possible $x$ -values, then think about what the question is asking you to choose.

Desmos Guide

Graph the line

In Desmos, type the equation of the line as $y = 5 - 2x$ (this is the same as $2x + y = 5$ solved for $y$ ).

Graph the circle

On a new line in Desmos, type the circle equation $x^2 + y^2 = 10$ .

Find the intersection points

Look for the intersection points between the line and the circle. Click each intersection and note the $x$ -coordinates. The correct response is the larger of these $x$ -values.

Step-by-step Explanation

Solve the linear equation for one variable

From the first equation

2x + y = 5,

solve for $y$ in terms of $x$ :

y = 5 - 2x.

Now you can substitute this expression for $y$ into the second equation.

Substitute into the circle equation and simplify

Substitute $y = 5 - 2x$ into the second equation $x^2 + y^2 = 10$ :

x^2 + (5 - 2x)^2 = 10.

Expand and simplify:

\begin{aligned} x^2 + (5 - 2x)^2 &= x^2 + (25 - 20x + 4x^2) \\ &= x^2 + 25 - 20x + 4x^2 \\ &= 5x^2 - 20x + 25. \end{aligned}

Set this equal to 10 and move all terms to one side:

\begin{aligned} 5x^2 - 20x + 25 &= 10 \\ 5x^2 - 20x + 15 &= 0. \end{aligned}

Divide everything by 5 to simplify:

x^2 - 4x + 3 = 0.

This is a quadratic equation in $x$ .

Solve the quadratic and pick the greater value

Factor the quadratic:

x^2 - 4x + 3 = (x - 1)(x - 3) = 0.

So the possible $x$ -values are $x = 1$ and $x = 3$ .

The question asks for the greater possible value of $x$ , so the correct answer is $3$ .

Strategy

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Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

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Desmos Guide

Step-by-step Explanation