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

Cevap: 7

Cevap

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

Adım Adım Çözüm

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.

Anahtar Kavram

Nonlinear Systems of Equations

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru