The path of a particle in the standard (x, y) coordinate plane is described by the linear … — ACT

The path of a particle in the standard $(x, y)$ coordinate plane is described by the linear equation $3x - y = 2$ , and the path of another particle is described by the quadratic equation $y = x^2 - x - 7$ . If the two paths intersect at two locations, what is the sum of the $y$ -coordinates of these intersection points?

Answer: 8

Answer

The sum of the

y

-coordinates of the intersection points is 8.

The correct answer is 8. Solving the system by setting

x^2 - x - 7 = 3x - 2

results in the quadratic equation

x^2 - 4x - 5 = 0

. Factoring gives

(x-5)(x+1) = 0

, which yields intersection

x

-coordinates of

x = 5

and

x = -1

. Substituting these back into the linear equation gives

y

-coordinates of

13

and

-5

. Adding these values together yields

13 + (-5) = 8

Step-by-Step Solution

Express the linear equation in terms of

y

y = 3x - 2

This allows for direct substitution into the quadratic equation.

Equate the linear and quadratic expressions to solve for the

x

-coordinates of the intersection points.

x^2 - x - 7 = 3x - 2

Intersection points share the same coordinates for both equations.

Set the quadratic equation to zero.

x^2 - 4x - 5 = 0

This puts the equation in standard form so it can be solved by factoring.

Factor the quadratic equation.

(x - 5)(x + 1) = 0

, so

x = 5

x = -1

Factoring determines the

x

-coordinates of the intersection points.

Substitute the

x

-values into the linear equation to determine the

y

-coordinates.

For

x = 5

y = 13

. For

x = -1

y = -5

Finding the

y

-coordinates is necessary to compute their sum.

Add the

y

-coordinates together.

13 + (-5) = 8

The question asks for the sum of the

y

-coordinates of the intersection points.

Key Concept

Solving systems of linear and quadratic equations by substitution and factoring

Alternative Method

We can use Vieta's formulas to find the sum of the

y

-coordinates without calculating each individual coordinate. The sum of the

y

-coordinates is

y_1 + y_2 = (3x_1 - 2) + (3x_2 - 2) = 3(x_1 + x_2) - 4

. Since

x_1

and

x_2

are the roots of

x^2 - 4x - 5 = 0

, Vieta's formulas state that the sum of the roots is

x_1 + x_2 = -\frac{-4}{1} = 4

. Substituting this value into our sum expression yields

3(4) - 4 = 12 - 4 = 8

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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