If (x_1, y_1) and (x_2, y_2) are the real solution pairs to the simultaneous equations x -… — JAMB UTME

If $(x_1, y_1)$ and $(x_2, y_2)$ are the real solution pairs to the simultaneous equations $x - 2y = 1$ and $x^2 - xy + y^2 = 7$ , what is the value of $x_1 x_2 + y_1 y_2$ ?

A
$-9$
$-11$ Answer
C
$-7$
D
$-1$

Answer

The value of

x_1 x_2 + y_1 y_2

-11

Substituting

x = 2y + 1

into

x^2 - xy + y^2 = 7

gives

3y^2 + 3y - 6 = 0

, which reduces to

y^2 + y - 2 = 0

. The roots are

y = 1

and

y = -2

. Substituting back into the linear expression yields corresponding values

x = 3

and

x = -3

. The two solution pairs are

(3, 1)

and

(-3, -2)

. Calculating

x_1 x_2 + y_1 y_2 = (3)(-3) + (1)(-2)

yields

-11

Step-by-Step Solution

Express

x

in terms of

y

from the linear equation.

x = 2y + 1

Isolating variable

x

allows direct substitution into the quadratic equation.

Substitute

x = 2y + 1

into the quadratic equation

x^2 - xy + y^2 = 7

(2y + 1)^2 - (2y + 1)y + y^2 = 7 \implies 3y^2 + 3y - 6 = 0

This simplifies the system to a single quadratic equation in

y

Solve the quadratic equation

3y^2 + 3y - 6 = 0

y^2 + y - 2 = 0 \implies (y + 2)(y - 1) = 0

, giving

y_1 = 1

and

y_2 = -2

Factoring determines the two possible

y

-coordinates.

Determine corresponding

x

-coordinates for each

y

-value.

For

y_1 = 1

x_1 = 2(1) + 1 = 3 \implies (3, 1)

. For

y_2 = -2

x_2 = 2(-2) + 1 = -3 \implies (-3, -2)

Substituting each

y

into

x = 2y + 1

yields the complete solution pairs.

Compute

x_1 x_2 + y_1 y_2

(3)(-3) + (1)(-2) = -9 - 2 = -11

Evaluates the targeted algebraic expression.

Key Concept

Solving simultaneous linear and quadratic equations using algebraic substitution.

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