What is the sum of the y-values of the solution pairs that satisfy the simultaneous equati… — JAMB UTME

What is the sum of the $y$ -values of the solution pairs that satisfy the simultaneous equations $y - x = 2$ and $y = x^2 - 3x + 5$ ?

A
$4$
B
$6$
$8$ Answer
D
$15$

Answer

The sum of the

y

-values is 8.

Rearranging the linear equation gives

y = x + 2

. Equating this to the quadratic expression gives

x + 2 = x^2 - 3x + 5

, which simplifies to

x^2 - 4x + 3 = 0

. Solving for

x

yields

x = 1

and

x = 3

. Substituting these into

y = x + 2

gives

y = 3

and

y = 5

. The sum of these

y

-values is

3 + 5 = 8

Step-by-Step Solution

Express

y

in terms of

x

from the linear equation.

y = x + 2

Isolating

y

facilitates substitution into the quadratic equation.

Substitute

y = x + 2

into the quadratic equation and rearrange into standard form.

x + 2 = x^2 - 3x + 5 \implies x^2 - 4x + 3 = 0

Setting the two expressions for

y

equal to each other forms a single quadratic equation in

x

Solve the quadratic equation for

x

(x - 1)(x - 3) = 0 \implies x_1 = 1, x_2 = 3

Factoring gives the

x

-coordinates of the intersection points.

Calculate the corresponding

y

-values using

y = x + 2

y_1 = 1 + 2 = 3

and

y_2 = 3 + 2 = 5

Each

x

-value must be substituted into the linear equation to find its paired

y

-value.

Find the sum of the

y

-values.

y_1 + y_2 = 3 + 5 = 8

The question asks for the sum of the

y

-coordinates.

Key Concept

Simultaneous Linear and Quadratic Equations

Alternative Method

Using Vieta's formulas: once the quadratic equation for

x

is found as

x^2 - 4x + 3 = 0

, we know

x_1 + x_2 = 4

. Since

y_1 = x_1 + 2

and

y_2 = x_2 + 2

, the sum of the

y

-values is

y_1 + y_2 = (x_1 + 2) + (x_2 + 2) = (x_1 + x_2) + 4 = 4 + 4 = 8

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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