If (x, y) satisfies the simultaneous equations x + 2y = 5 and x^2 + y^2 = 10, what is the … — JAMB UTME

If $(x, y)$ satisfies the simultaneous equations $x + 2y = 5$ and $x^2 + y^2 = 10$ , what is the positive value of $x$ ?

Answer: 3

Answer

The positive value of

x

is 3.

Isolating

x

in the linear equation gives

x = 5 - 2y

. Substituting this expression into

x^2 + y^2 = 10

yields

(5 - 2y)^2 + y^2 = 10

. Expanding gives

25 - 20y + 4y^2 + y^2 = 10 \implies 5y^2 - 20y + 15 = 0

. Dividing all terms by

5

produces

y^2 - 4y + 3 = 0

, which factors as

(y - 1)(y - 3) = 0

, so

y = 1

y = 3

. Substituting these into

x = 5 - 2y

gives

x = 3

when

y = 1

and

x = -1

when

y = 3

. The positive value of

x

is 3.

Step-by-Step Solution

Express

x

from the linear equation

x = 5 - 2y

Isolating

x

allows substitution into the quadratic equation.

Substitute into the quadratic equation

(5 - 2y)^2 + y^2 = 10

Eliminates variable

x

to create a single-variable equation in

y

Expand and simplify

5y^2 - 20y + 15 = 0 \implies y^2 - 4y + 3 = 0

Transforms the equation into standard quadratic form for easy factorization.

Solve for

y

y = 1 \text{ or } y = 3

Factoring

(y - 1)(y - 3) = 0

yields the two possible values for

y

Determine corresponding

x

values and select the positive one

x = 3

(from

y = 1

)

Evaluating

x = 5 - 2y

gives

x = 3

and

x = -1

; the positive result requested is 3.

Key Concept

Solving simultaneous linear and quadratic equations by substitution

Estimated Time:1m 30s

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