Question

Solve the simultaneous equations
5x-y=5
2y-x²=11
YOU MUST SHOW YOUR WORKING

Answer 1

Correct Equations:-

`\sf 5x-y=5 \\ \sf 2y-x=11`

solution:-

`\sf 5x-y=5\dots\dots(1)`

`\sf -x+2y=11 \dots\dots (2)`

• We use elimination method
• By multiplying eq (1) by 2 and eq (2) by 1 we get

`{:}\longrightarrow`
`\sf 10x-2y=5\dots\dots(3)`

`{:}\longrightarrow`
`\sf -x+2y=11\dots\dots (4)`

______________________

• Add eq (3) and eq (4)

`{:}\longrightarrow`
`\sf 4x=16`

`{:}\longrightarrow`
`\sf x={\frac {16}{4}}`

`{:}\longrightarrow`
`\sf x=4`

• Substitute the value in eq (2)

`{:}\longrightarrow`
`\sf -4+2y=11`

`{:}\longrightarrow`
`\sf 2y=11+4`

`{:}\longrightarrow`
`\sf 2y=15`

`{:}\longrightarrow`
`\sf y={\frac {15}{2}}`

`\therefore`
`\sf (x,y)=(4,{\frac {5}{2}})`

Answer 2

(3, 10 ) and (7, 30 )

Explanation:

Given the 2 equations

5x - y = 5 → (1)

2y - x² = 11 → (2)

Rearrange (1) expressing y in terms of x

y = 5x - 5 → (3)

Substitute y = 5x - 5 into (2)

2(5x - 5) - x² = 11

10x - 10 - x² = 11 ← rearrange into standard form

x² - 10x + 21 = 0 ← in standard form

(x - 3)(x - 7) = 0 ← in factored form

Equate each factor to zero and solve for x

x - 3 = 0 ⇒ x = 3

x - 7 = 0 ⇒ x = 7

Substitute these values into (3) for corresponding values of y

x = 3 → y = 5(3) - 5 = 15 - 5 = 10 ⇒ (3, 10 )

x = 7 → y = 5(7) - 5 = 35 - 5 = 30 ⇒ (7, 30 )