A system of equations is shown below. y = x^2 - 2x y = 3 If (x, y) is a solution to the sy… — SAT

A system of equations is shown below.

y = x^2 - 2x

y = 3

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

To solve the system of equations, substitute the expression for

y

from the second equation into the first equation:

3 = x^2 - 2x

. Subtracting 3 from both sides results in the quadratic equation

x^2 - 2x - 3 = 0

. Factoring the quadratic yields

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

, which gives the possible values of

x

3

and

-1

. Since the problem specifies that

x > 0

, the value of

x

is 3. Given that

y = 3

, the value of

x + y

3 + 3 = 6

Substitute the value of

y

from the second equation into the first equation.

3 = x^2 - 2x

Since both equations are equal to

y

, their right-hand sides must be equal to each other.

Rearrange the equation to set it equal to zero.

x^2 - 2x - 3 = 0

This puts the equation into standard quadratic form so that it can be factored.

Factor the quadratic equation.

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

Finding factors of -3 that add up to -2 helps isolate the solutions for

x

Find the solutions for

x

and apply the given constraint.

x = 3

(since

x > 0

)

The factors give

x = 3

and

x = -1

. The constraint

x > 0

excludes

x = -1

Calculate the value of

x + y

3 + 3 = 6

Substitute the value of

x = 3

and the given value of

y = 3

to find the final sum.

Solving nonlinear systems of equations using substitution and solving quadratic equations by factoring.

Question