\begin{aligned} y + 11 &= x^2 \\ y - 3x &= 7 \end{aligned} If (x, y) is a solution to the … — SAT

\begin{aligned} y + 11 &= x^2 \\ y - 3x &= 7 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $x > 0$ , what is the value of $x + y$ ?

Answer: 31

Answer

The correct answer is 31.

Solving the system of equations by expressing

y

in terms of

x

from the linear equation gives

y = 3x + 7

. Substituting this expression for

y

in the first equation yields

(3x + 7) + 11 = x^2

, which simplifies to

x^2 - 3x - 18 = 0

. Factoring the quadratic expression gives

(x - 6)(x + 3) = 0

, so

x = 6

x = -3

. Since the problem specifies that

x > 0

, the

x

-value of the solution must be

6

. Substituting

x = 6

back into the linear equation gives

y = 3(6) + 7 = 25

. The sum of

x

and

y

6 + 25 = 31

Step-by-Step Solution

Express

y

in terms of

x

from the linear equation.

y = 3x + 7

This isolates the variable

y

to facilitate substitution.

Substitute the expression for

y

into the quadratic equation.

(3x + 7) + 11 = x^2

This eliminates the variable

y

, yielding a single equation in terms of

x

Rearrange the equation into standard quadratic form.

x^2 - 3x - 18 = 0

Standard form (

ax^2 + bx + c = 0

) is required to solve by factoring.

Factor the quadratic equation.

(x - 6)(x + 3) = 0

Finding factors helps determine the possible values of

x

Find the values of

x

and apply the positive constraint.

x = 6

The equation yields

x = 6

and

x = -3

. The condition

x > 0

restricts the solution to

x = 6

Calculate the corresponding value of

y

y = 25

Substituting

x = 6

into the linear equation gives

y = 3(6) + 7 = 25

Calculate the sum of

x

and

y

31

The problem asks for the value of

x + y

Key Concept

Solving systems of nonlinear equations algebraically using substitution and quadratic factoring.

Alternative Method

Alternatively, solve the linear equation for

x

to get

x = \frac{y - 7}{3}

and substitute this into the quadratic equation to solve for

y

first. This approach is more complex because it introduces fractional terms.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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