If x and y satisfy the simultaneous equations x - 2y = 1 and x^2 - xy - y^2 = 5, what is t… — JAMB UTME

If $x$ and $y$ satisfy the simultaneous equations $x - 2y = 1$ and $x^2 - xy - y^2 = 5$ , what is the positive value of $x$ ?

Answer: 3

The positive value of

x

is 3.

Substituting

x = 2y + 1

into the non-linear equation yields

(2y + 1)^2 - (2y + 1)y - y^2 = 5

, which simplifies to

y^2 + 3y - 4 = 0

. Solving this gives

y = 1

y = -4

. Substituting

y = 1

into

x = 2y + 1

gives

x = 3

, which is the positive value of

x

Make

x

the subject of the linear equation

x = 2y + 1

Substitution is the standard method for solving linear-quadratic simultaneous systems.

Substitute

x = 2y + 1

into

x^2 - xy - y^2 = 5

(2y + 1)^2 - (2y + 1)y - y^2 = 5

Eliminates

x

to create a single quadratic equation in terms of

y

Expand and simplify to standard quadratic form

y^2 + 3y - 4 = 0

Simplifying algebraic expressions allows factorization.

Solve for

y

by factoring

y = 1

y = -4

Factors of

-4

that sum to

3

are

+4

and

-1

Calculate corresponding values for

x

x = 3

when

y = 1

, and

x = -7

when

y = -4

Substitute

y

back into the linear expression for

x

Select the positive value of

x

The question explicitly asks for the positive value of

x

Simultaneous Linear and Quadratic Equations

Question