A linear equation and a quadratic equation form a system, as shown. y = x^2 + 5 y = 3x + 5… — SAT

A linear equation and a quadratic equation form a system, as shown.

y = x^2 + 5

y = 3x + 5

If the ordered pair $(x, y)$ is a solution to the system where $x$ is positive, what is the value of $x$ ?

Answer: 3

The value of

x

is 3.

Equating the two expressions for

y

gives the equation

x^2 + 5 = 3x + 5

. Subtracting 5 from both sides results in

x^2 = 3x

. Subtracting

3x

from both sides gives the quadratic equation

x^2 - 3x = 0

, which can be factored as

x(x - 3) = 0

. This yields two solutions for

x

0

and

3

. Since the question specifies that

x

is positive, the correct value is 3.

Equate the equations

x^2 + 5 = 3x + 5

Since both equations are solved for

y

, we can substitute the quadratic expression into the linear equation.

Simplify the equation

x^2 - 3x = 0

Subtracting 5 from both sides and then subtracting

3x

from both sides collects all terms on one side.

Factor and solve

x(x - 3) = 0

, so

x = 0

x = 3

Factoring out the greatest common factor,

x

, allows us to apply the zero product property to find the individual roots.

Apply the constraint

x = 3

The problem states that

x

must be positive, which excludes the solution

x = 0

Solving a system consisting of a linear equation and a quadratic equation by substitution.

Question