Consider the system of simultaneous equations x - 3y = 2 and x^2 - 2xy - 4y^2 = 19. What i… — JAMB UTME

Consider the system of simultaneous equations $x - 3y = 2$ and $x^2 - 2xy - 4y^2 = 19$ . What is the sum of the $x$ -values of the real solution pairs $(x, y)$ ?

$28$ Answer
B
$8$
C
$20$
D
$-20$

Answer

The sum of the

x

-values of the real solution pairs is

28

Rearranging the linear equation gives

x = 3y + 2

. Substituting this into

x^2 - 2xy - 4y^2 = 19

results in

(3y + 2)^2 - 2(3y + 2)y - 4y^2 = 19

, which simplifies to

y^2 - 8y + 15 = 0

. The roots are

y = 5

and

y = 3

. Substituting these back into

x = 3y + 2

yields

x = 17

and

x = 11

. Their sum is

17 + 11 = 28

Step-by-Step Solution

Express

x

in terms of

y

using the linear equation.

x = 3y + 2

Isolation of

x

allows direct substitution into the quadratic equation.

Substitute

x = 3y + 2

into the quadratic equation

x^2 - 2xy - 4y^2 = 19

(3y + 2)^2 - 2(3y + 2)y - 4y^2 = 19

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

y

Expand and simplify the quadratic equation.

(9y^2 + 12y + 4) - (6y^2 + 4y) - 4y^2 = 19 \implies -y^2 + 8y + 4 = 19 \implies y^2 - 8y + 15 = 0

Putting the quadratic expression into standard form

ay^2 + by + c = 0

facilitates finding its roots.

Solve the quadratic equation

y^2 - 8y + 15 = 0

for

y

(y - 5)(y - 3) = 0 \implies y_1 = 5, y_2 = 3

Factoring determines the ordinate values for the solution pairs.

Calculate the corresponding

x

-values and find their sum.

For

y_1 = 5

x_1 = 3(5) + 2 = 17

.
For

y_2 = 3

x_2 = 3(3) + 2 = 11

.
Sum =

17 + 11 = 28

Plugging the

y

-values back into

x = 3y + 2

gives the abscissas, which are added to answer the question.

Key Concept

Solving simultaneous linear and quadratic equations by algebraic substitution

Estimated Time:2m 0s

Question

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