If (x_1, y_1) and (x_2, y_2) are the real solution pairs to the simultaneous equations 3x … — JAMB UTME

If $(x_1, y_1)$ and $(x_2, y_2)$ are the real solution pairs to the simultaneous equations $3x - y = 5$ and $x^2 + 2xy - y^2 = 7$ , with $x_1 < x_2$ , what is the value of $y_2 - y_1$ ?

18Answer
B
6
C
20
D
14

Answer

Expressing

y

3x - 5

and substituting into the quadratic equation yields

x^2 - 10x + 16 = 0

. Solving gives

x_1 = 2

and

x_2 = 8

. Evaluating

y = 3x - 5

for both values gives

y_1 = 1

and

y_2 = 19

. The difference

y_2 - y_1 = 19 - 1 = 18

Step-by-Step Solution

Express

y

in terms of

x

from the linear equation

y = 3x - 5

Isolation of

y

facilitates substitution into the non-linear equation.

Substitute

y = 3x - 5

into the quadratic equation

x^2 + 2xy - y^2 = 7

x^2 + 2x(3x - 5) - (3x - 5)^2 = 7

Reduces the system to a single quadratic equation in terms of

x

Expand and simplify the algebraic expression

x^2 + 6x^2 - 10x - (9x^2 - 30x + 25) = 7 \implies -2x^2 + 20x - 32 = 0 \implies x^2 - 10x + 16 = 0

Simplifies the equation to standard quadratic form

ax^2 + bx + c = 0

Solve the quadratic equation

x^2 - 10x + 16 = 0

by factorization

(x - 2)(x - 8) = 0 \implies x_1 = 2

and

x_2 = 8

Identifies the two roots with

x_1 < x_2

as specified by the condition.

Determine the corresponding

y

-values using

y = 3x - 5

y_1 = 3(2) - 5 = 1

and

y_2 = 3(8) - 5 = 19

Obtains the complete coordinate solution pairs

(2, 1)

and

(8, 19)

Calculate the required difference

y_2 - y_1

y_2 - y_1 = 19 - 1 = 18

Computes the final required target value.

Key Concept

Solving Simultaneous Linear and Quadratic Equations by Substitution

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