If (x, y) is a pair of positive integers satisfying the simultaneous equations x + 2y = 7 … — JAMB UTME

If $(x, y)$ is a pair of positive integers satisfying the simultaneous equations $x + 2y = 7$ and $x^2 + y^2 = 13$ , what is the value of $2x - y$ ?

Cevap: 4

The value of

2x - y

is 4.

Substituting

x = 7 - 2y

into

x^2 + y^2 = 13

yields the quadratic

5y^2 - 28y + 36 = 0

, which factors as

(5y - 18)(y - 2) = 0

. Since

x

and

y

must both be positive integers, the valid solution pair is

(x, y) = (3, 2)

. Evaluating

2x - y

gives

2(3) - 2 = 4

Express

x

in terms of

y

using the linear equation

x + 2y = 7

x = 7 - 2y

Isolating

x

allows substitution into the quadratic equation.

Substitute

x = 7 - 2y

into

x^2 + y^2 = 13

and expand.

(7 - 2y)^2 + y^2 = 13 \implies 49 - 28y + 4y^2 + y^2 = 13

This reduces the system to a single quadratic equation in terms of

y

Rearrange and solve the resulting quadratic equation

5y^2 - 28y + 36 = 0

(5y - 18)(y - 2) = 0 \implies y = 2

y = 3.6

Factoring provides the possible values for

y

Identify the integer solution pair and compute the target expression

2x - y

Since

y

must be a positive integer,

y = 2

and

x = 3

. Therefore,

2x - y = 2(3) - 2 = 4

The non-integer solution is rejected based on the positive integer constraint.

Simultaneous Linear and Quadratic Equations

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