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

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SAT Nonlinear Equations & Systems Question 189 | Free SAT Prep | Aniko

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Question 189·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The system of equations is given by

\begin{aligned} x + y &= 10 \\ x^2 + y^2 &= 58 \end{aligned}

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

When you see a system with a sum like $x + y = \text{(number)}$ and a sum of squares like $x^2 + y^2 = \text{(number)}$ , think of the identity $(x + y)^2 = x^2 + 2xy + y^2$ to quickly find the product $xy$ . Once you know both $x + y$ and $xy$ , build a quadratic whose roots are $x$ and $y$ using $t^2 - (x + y)t + xy = 0$ , then solve it by factoring or the quadratic formula. This approach avoids messy substitution and is fast and reliable under time pressure.

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Connect the two equations

You know $x + y = 10$ and $x^2 + y^2 = 58$ . How can you use the first equation to create an expression that also involves $x^2$ and $y^2$ ?

Use a key algebra identity

Remember that $(x + y)^2$ is not just $x^2 + y^2$ . Think about the full expansion of $(x + y)^2$ and how it includes a term with $xy$ .

Turn it into a quadratic

Once you know $x + y$ and $xy$ , recall how to build a quadratic equation whose roots are $x$ and $y$ using their sum and product.

Desmos Guide

Graph the line from the first equation

In Desmos, enter the first equation as x + y = 10. This graphs a straight line representing all pairs $(x, y)$ that satisfy $x + y = 10$ .

Graph the curve from the second equation

In a new line, enter x^2 + y^2 = 58. This graphs a circle (all points whose coordinates satisfy $x^2 + y^2 = 58$ ).

Find the intersection points

Look for the intersection points of the line and the circle. The $x$ -coordinates of these intersection points are the possible values of $x$ that satisfy both equations. Read off either of those $x$ -values from the graph.

Step-by-step Explanation

Square the sum of x and y

From the first equation, you know that

x + y = 10.

Square both sides to relate this to the second equation:

(x + y)^2 = 10^2 = 100.

Relate (x + y)^2 to x^2 + y^2 and find xy

Use the algebra identity

(x + y)^2 = x^2 + 2xy + y^2.

Substitute into the result from Step 1:

100 = x^2 + 2xy + y^2.

From the second equation, you know

x^2 + y^2 = 58.

Replace $x^2 + y^2$ with 58:

100 = 58 + 2xy.

Subtract 58 from both sides:

42 = 2xy,

xy = 21.

Use sum and product to form a quadratic

Now you know two facts about $x$ and $y$ :

Their sum is $x + y = 10$ .
Their product is $xy = 21$ .

That means $x$ and $y$ are the two roots of a quadratic equation whose sum of roots is 10 and whose product of roots is 21. A quadratic with roots $x$ and $y$ is

t^2 - (\text{sum})t + (\text{product}) = 0,

so here it becomes

t^2 - 10t + 21 = 0.

Solve the quadratic to find possible x-values

Factor the quadratic

t^2 - 10t + 21 = 0.

Look for two numbers that multiply to 21 and add to 10: they are 3 and 7. So the equation factors as

(t - 3)(t - 7) = 0.

This gives

t = 3 \quad \text{or} \quad t = 7.

So the possible values for $x$ (and $y$ ) are 3 and 7. The problem asks for one possible value of $x$ , so a correct response is 7.

Strategy

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Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation