Given the simultaneous equations 2x + y = 5 and x^2 + xy - y^2 = -5, let (x_1, y_1) and (x… — JAMB UTME

Given the simultaneous equations $2x + y = 5$ and $x^2 + xy - y^2 = -5$ , let $(x_1, y_1)$ and $(x_2, y_2)$ be the real solution pairs. What is the value of $x_1 x_2 + y_1 y_2$ ?

$-5$ Answer
B
$13$
C
$5$
D
$95$

Answer

-5

Substituting

y = 5 - 2x

into

x^2 + xy - y^2 = -5

yields

-5x^2 + 25x - 20 = 0

, which simplifies to

x^2 - 5x + 4 = 0

. The roots are

x_1 = 1

and

x_2 = 4

, giving corresponding

y

-values

y_1 = 3

and

y_2 = -3

. Evaluating

x_1 x_2 + y_1 y_2

gives

(1)(4) + (3)(-3) = 4 - 9 = -5

Step-by-Step Solution

Express

y

in terms of

x

using the linear equation.

y = 5 - 2x

Substitution method requires isolating one variable from the linear equation.

Substitute

y = 5 - 2x

into the quadratic equation

x^2 + xy - y^2 = -5

x^2 + x(5 - 2x) - (5 - 2x)^2 = -5

Form a single quadratic equation in terms of

x

Expand and simplify the quadratic equation.

x^2 + 5x - 2x^2 - (25 - 20x + 4x^2) = -5 \implies -5x^2 + 25x - 20 = 0 \implies x^2 - 5x + 4 = 0

Convert the equation to standard quadratic form

ax^2 + bx + c = 0

Factorize the quadratic equation to find the values of

x

(x - 1)(x - 4) = 0 \implies x_1 = 1, x_2 = 4

Obtain the two roots for

x

Find the corresponding

y

-values using

y = 5 - 2x

For

x_1 = 1

y_1 = 5 - 2(1) = 3

. For

x_2 = 4

y_2 = 5 - 2(4) = -3

. Solution pairs are

(1, 3)

and

(4, -3)

Calculate corresponding coordinate values for each root.

Compute the required expression

x_1 x_2 + y_1 y_2

x_1 x_2 + y_1 y_2 = (1)(4) + (3)(-3) = 4 - 9 = -5

Perform the final calculation requested in the stem.

Key Concept

Simultaneous Linear and Quadratic Equations

Question

Answer

Step-by-Step Solution

Key Concept

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