Question

if 3x+4y=11 and xy=2 solve simultaneously​

Answer 1

Answer:

(1, 2 ) and ($\frac{8}{3}$, $\frac{3}{4}$ )

Step-by-step explanation:

3x + 4y = 11 → (1)

xy = 2 ( divide both sides by y , y ≠ 0 )

x = $\frac{2}{y}$ → (2)

Substitute x = $\frac{2}{y}$ into (1)

3($\frac{2}{y}$ ) + 4y = 11

$\frac{6}{y}$ + 4y = 11 ( multiply through by y )

6 + 4y² = 11y ( subtract 11y from both sides )

4y² - 11y + 6 = 0

Consider the factors of the product of the y² term and the constant term which sum to give the coefficient of the y- term

product = 4 × 6 = 24 and sum = - 11

The factors are - 8 and - 3

Use these factors to split the y- term

4y² - 8y - 3y + 6 = 0 ( factor first/second and third/fourth terms )

4y(y - 2) - 3(y - 2) = 0 ← factor out (y - 2) from each term

(y - 2)(4y - 3) = 0

Equate each factor to zero and solve for y

4y - 3 = 0 ⇒ 4y = 3 ⇒ y = $\frac{3}{4}$

y - 2 = 0 ⇒ y = 2

Substitute these values into (2)

y = 2 : x = $\frac{2}{2}$ = 1 ⇒ (1, 2 )

y = $\frac{3}{4}$ : x = $\frac{2}{\frac{3}{4} }$ = 2 × $\frac{4}{3}$ = $\frac{8}{3}$ ⇒ ( $\frac{8}{3}$, $\frac{3}{4}$ )