A system of equations consists of the linear equation y = 2x + 1 and the quadratic equatio… — ACT

A system of equations consists of the linear equation $y = 2x + 1$ and the quadratic equation $y = x^2 - 2$ . If $(x, y)$ is a solution to this system such that $x > 0$ , what is the value of $y$ ?

Answer: 7

Answer

The correct value of

y

is 7.

Substituting

x = 3

into either equation yields the

y

-value of 7.

Step-by-Step Solution

Set the two expressions for

y

equal to each other.

x^2 - 2 = 2x + 1

Since both equations define

y

in terms of

x

, equating them allows us to find the

x

-coordinates of the intersection points.

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

x^2 - 2x - 3 = 0

Subtracting

2x

and

1

from both sides collects all terms on one side of the equation.

Factor the quadratic equation.

(x - 3)(x + 1) = 0

Finding two numbers that multiply to

-3

and add to

-2

gives

-3

and

1

, allowing the quadratic to be factored.

Solve for

x

and apply the constraint

x > 0

x = 3

The factored equation yields solutions of

x = 3

and

x = -1

. The constraint that

x

must be greater than zero means we select

x = 3

Substitute the

x

-value back into one of the original equations to solve for

y

y = 7

Plugging

x = 3

into the linear equation

y = 2x + 1

yields

y = 2(3) + 1 = 7

. Substituting into

y = x^2 - 2

yields

y = 3^2 - 2 = 7

, which confirms the solution.

Key Concept

Solving a system of linear and quadratic equations using substitution.

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