If (u, v) is a pair of real numbers satisfying the simultaneous equations u - 2v = 1 and u… — JAMB UTME

If $(u, v)$ is a pair of real numbers satisfying the simultaneous equations $u - 2v = 1$ and $u^2 - 3v^2 = 13$ , what is the sum of all possible values of $u$ ?

$-6$ Answer
B
$-4$
C
$6$
D
$-8$

Answer

The sum of all possible values of

u

-6

From the linear equation

u - 2v = 1

, we express

u

u = 2v + 1

. Substituting this into the quadratic equation

u^2 - 3v^2 = 13

gives

(2v + 1)^2 - 3v^2 = 13

, which expands and simplifies to

v^2 + 4v - 12 = 0

. Factoring gives

(v + 6)(v - 2) = 0

, so

v = 2

v = -6

. Substituting these back into

u = 2v + 1

yields

u = 5

(for

v = 2

) and

u = -11

(for

v = -6

). The sum of all possible values of

u

5 + (-11) = -6

Step-by-Step Solution

Express

u

in terms of

v

using the linear equation.

u = 2v + 1

Isolating

u

allows for straightforward substitution into the quadratic equation.

Substitute

u = 2v + 1

into the quadratic equation

u^2 - 3v^2 = 13

and expand.

(2v + 1)^2 - 3v^2 = 13 \implies (4v^2 + 4v + 1) - 3v^2 = 13 \implies v^2 + 4v - 12 = 0

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

v

Solve the quadratic equation for

v

by factorisation.

(v + 6)(v - 2) = 0 \implies v = 2 \text{ or } v = -6

Finding the roots of the quadratic gives the

v

-coordinates of the solution pairs.

Calculate the corresponding values of

u

using

u = 2v + 1

For

v = 2

u = 2(2) + 1 = 5

.
For

v = -6

u = 2(-6) + 1 = -11

Each

v

value corresponds to a specific

u

value in the solution pairs.

Compute the sum of all possible values of

u

5 + (-11) = -6

The question asks specifically for the sum of the

u

-values.

Key Concept

Solving simultaneous linear and quadratic equations using substitution

Question

Answer

Step-by-Step Solution

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