If (x, y) is a solution to the system of equations below and x > 0, what is the value of x… — SAT

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

y = x^2

y = x + 6

Answer: 3

The correct answer is

3

Substituting

y = x^2

into

y = x + 6

gives the quadratic equation

x^2 - x - 6 = 0

. Factoring this expression yields

(x - 3)(x + 2) = 0

, which gives solutions of

x = 3

and

x = -2

. Since the system requires

x > 0

, the only valid solution is

3

Substitute the expression for

y

from the first equation into the second equation.

x^2 = x + 6

To eliminate the variable

y

and solve for

x

directly.

Subtract

x

and

6

from both sides to write the quadratic equation in standard form.

x^2 - x - 6 = 0

Setting the quadratic expression equal to zero allows it to be factored.

Factor the quadratic trinomial.

(x - 3)(x + 2) = 0

Finding factors whose product is

-6

and whose sum is

-1

helps find the roots.

Solve for

x

and apply the constraint

x > 0

x = 3

The equation has solutions

x = 3

and

x = -2

. Because

x

must be greater than

0

, we discard the negative solution.

Solving a system of nonlinear equations by substitution and factoring the resulting quadratic equation.

Question