If (x, y) satisfies the simultaneous equations x - y = 1 and x^2 + y^2 = 25, what is the v… — JAMB UTME

If $(x, y)$ satisfies the simultaneous equations $x - y = 1$ and $x^2 + y^2 = 25$ , what is the value of the product $xy$ ?

Answer: 12

Expanding

(x - y)^2

yields

x^2 - 2xy + y^2

. Substituting the given values

x - y = 1

and

x^2 + y^2 = 25

into this identity gives

1 = 25 - 2xy

. Rearranging yields

2xy = 24

, which solves to

xy = 12

. Alternatively, solving by substitution gives solution pairs

(4, 3)

and

(-3, -4)

, both yielding a product of

12

Apply the algebraic expansion identity

(x - y)^2 = x^2 + y^2 - 2xy

Connects the difference of terms, the sum of their squares, and their product.

Substitute the values given in the system of equations

1^2 = 25 - 2xy

Replaces

x - y

with 1 and

x^2 + y^2

with 25.

Isolate and calculate the product

xy

2xy = 24 \implies xy = 12

Simplifies

1 = 25 - 2xy

to find the exact numerical value of

xy

Simultaneous Linear and Quadratic Equations

Question