The system of equations below consists of a linear equation and a quadratic equation: \beg… — SAT

The system of equations below consists of a linear equation and a quadratic equation:

\begin{aligned} y - 2x &= 5 \\ y &= x^2 - 3x - 1 \end{aligned}

Let

(x_1, y_1)

and

(x_2, y_2)

represent the two distinct real coordinate points where the graphs of these equations intersect. If

y_1 > y_2

, what is the value of

x_1 - x_2

Answer: 7

Answer

Solving the system algebraically by expressing

y

2x + 5

and substituting it into the quadratic equation leads to

x^2 - 5x - 6 = 0

. Factoring this equation yields

x = 6

and

x = -1

. Evaluating these values in the linear equation gives the coordinates

(6, 17)

and

(-1, 3)

. Since

17 > 3

, the point

(6, 17)

corresponds to

(x_1, y_1)

and

(-1, 3)

corresponds to

(x_2, y_2)

. The difference

x_1 - x_2

is equal to

6 - (-1) = 7

Step-by-Step Solution

Solve for

y

in the linear equation to express it in terms of

x

y = 2x + 5

This allows for substitution into the quadratic equation to eliminate one variable.

Substitute

2x + 5

for

y

in the quadratic equation.

2x + 5 = x^2 - 3x - 1

To create a single-variable quadratic equation in terms of

x

Rearrange the equation by subtracting

2x

and

5

from both sides to set it equal to zero.

x^2 - 5x - 6 = 0

Setting the quadratic equation to zero is the standard first step to find its roots.

Factor the quadratic trinomial.

(x - 6)(x + 1) = 0

, which gives

x = 6

and

x = -1

Factoring finds the

x

-coordinates of the points of intersection.

Substitute the

x

-values back into the linear equation

y = 2x + 5

to determine the corresponding

y

-values.

For

x = 6

y = 2(6) + 5 = 17

. For

x = -1

y = 2(-1) + 5 = 3

. The intersection points are

(6, 17)

and

(-1, 3)

This defines the full coordinate pairs of the system's solutions.

Apply the condition

y_1 > y_2

to identify

(x_1, y_1)

and

(x_2, y_2)

Since

17 > 3

, the first solution is

(x_1, y_1) = (6, 17)

and the second solution is

(x_2, y_2) = (-1, 3)

This aligns the coordinate values with the variables defined in the problem constraint.

Calculate the value of

x_1 - x_2

6 - (-1) = 7

Subtracting a negative value is equivalent to adding its absolute value, yielding the final required answer.

Key Concept

Nonlinear Systems of Equations

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