Given the simultaneous equations x - y = 1 and x^2 + y^2 = 25, where x and y are both posi… — JAMB UTME

Given the simultaneous equations $x - y = 1$ and $x^2 + y^2 = 25$ , where $x$ and $y$ are both positive real numbers, what is the value of $x + y$ ?

The value of

x + y

7

Expressing

x

y + 1

and substituting into

x^2 + y^2 = 25

yields

2y^2 + 2y - 24 = 0

. Factoring gives

y = 3

(rejecting

y = -4

y > 0

). Substituting

y = 3

back gives

x = 4

. Adding these values together yields

x + y = 7

Express

x

in terms of

y

using the linear equation.

x = y + 1

Rearranging

x - y = 1

allows substitution into the quadratic equation.

Substitute

x = y + 1

into the quadratic equation

x^2 + y^2 = 25

(y + 1)^2 + y^2 = 25 \implies y^2 + 2y + 1 + y^2 = 25 \implies 2y^2 + 2y - 24 = 0

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

y

Solve the quadratic equation for

y

y^2 + y - 12 = 0 \implies (y + 4)(y - 3) = 0 \implies y = 3

y = -4

Dividing by

2

simplifies the equation, and factoring gives the potential roots for

y

Select the positive value of

y

and calculate

x

and

x + y

Since

y > 0

y = 3

. Then

x = 3 + 1 = 4

, so

x + y = 4 + 3 = 7

The question specifies that both

x

and

y

are positive real numbers.

Solving Simultaneous Linear and Quadratic Equations by Substitution

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