If 2x + y = 7 and x^2 + xy = 6, what are the possible values of x? — JAMB UTME

If $2x + y = 7$ and $x^2 + xy = 6$ , what are the possible values of $x$ ?

The possible values of

x

are

1

6

From the linear equation

2x + y = 7

, we get

y = 7 - 2x

. Substituting this into

x^2 + xy = 6

gives

x^2 + x(7 - 2x) = 6

, which simplifies to

-x^2 + 7x = 6

, or

x^2 - 7x + 6 = 0

. Factoring gives

(x - 1)(x - 6) = 0

, leading to

x = 1

x = 6

Express

y

in terms of

x

using the linear equation.

y = 7 - 2x

Isolation of one variable allows substitution into the non-linear equation.

Substitute

y = 7 - 2x

into the second equation

x^2 + xy = 6

x^2 + x(7 - 2x) = 6 \implies x^2 + 7x - 2x^2 = 6 \implies -x^2 + 7x - 6 = 0

This reduces the system to a single quadratic equation in

x

Rearrange into standard quadratic form

ax^2 + bx + c = 0

x^2 - 7x + 6 = 0

Standard quadratic form enables easy factorization.

Factor the quadratic equation and solve for

x

(x - 1)(x - 6) = 0 \implies x = 1 \text{ or } x = 6

Setting each factor to zero yields the values of

x

Solving simultaneous linear and quadratic equations by substitution

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