What are the values of y for the real solution pairs (x, y) that satisfy the simultaneous … — JAMB UTME

What are the values of $y$ for the real solution pairs $(x, y)$ that satisfy the simultaneous equations $y - 3x = 2$ and $y = x^2 - x + 5$ ?

5

11

Substituting

y = 3x + 2

into the quadratic equation

y = x^2 - x + 5

gives

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

, which simplifies to

x^2 - 4x + 3 = 0

. Solving this yields

x = 1

and

x = 3

. Substituting these into the linear equation gives

y = 3(1) + 2 = 5

and

y = 3(3) + 2 = 11

. Therefore, the possible values of

y

are

5

11

Express

y

in terms of

x

from the linear equation

y = 3x + 2

Linear equations can easily be substituted into quadratic equations.

Equate the linear expression for

y

to the quadratic equation

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

Both expressions represent

y

Rearrange into standard quadratic form

ax^2 + bx + c = 0

x^2 - 4x + 3 = 0

Subtract

3x

and

2

from both sides.

Factorize and solve for

x

(x - 1)(x - 3) = 0 \implies x = 1 \text{ or } x = 3

Finding the roots of the quadratic equation gives the

x

-coordinates of the solution pairs.

Substitute each

x

-value back into

y = 3x + 2

to find

y

When

x = 1

y = 3(1) + 2 = 5

. When

x = 3

y = 3(3) + 2 = 11

The question specifically asks for the values of

y

Simultaneous Linear and Quadratic Equations

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