Question

j 149b solve the following simultaneous equation

(2) \\(\

$$\begin{cases} xy = 12 \\\\ x^2 + y^2 = 25 \\end{cases}$$

\\)

Explanation:

Step1: Express the sum of squares using algebraic identity

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

Step2: Substitute the given values into the identity

$$(x + y)^2 = 25 + 2(12) = 49$$

Step3: Solve for the sum of the variables

$$x + y = \pm 7$$

Step4: Express the difference of squares using algebraic identity

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

Step5: Substitute the given values into the identity

$$(x - y)^2 = 25 - 2(12) = 1$$

Step6: Solve for the difference of the variables

$$x - y = \pm 1$$

Step7: Solve the system for positive sum and positive difference

$$
LATEXBLOCK0
\implies (x, y) = (4, 3)$$

Step8: Solve the system for positive sum and negative difference

$$
LATEXBLOCK1
\implies (x, y) = (3, 4)$$

Step9: Solve the system for negative sum and positive difference

$$
LATEXBLOCK2
\implies (x, y) = (-3, -4)$$

Step10: Solve the system for negative sum and negative difference

$$
LATEXBLOCK3
\implies (x, y) = (-4, -3)$$

Answer:

\((x, y) = (3, 4)\), \((4, 3)\), \((-3, -4)\), or \((-4, -3)\)