Question

Please can someone help!​

Answer 1

(x, y) ≈ (-0.7, -1.2) or (128.7, 13.2)

Explanation:

You want the solution to the simultaneous equations ...

• x -9y = 10
• 3y² = 4x +7

a) Quadratic

Solving the first equation for x, we can substitute into the second equation to get a quadratic in y.

x = 9y +10 . . . . . . . add 9y to the first equation

3y² = 4(9y +10) +7 . . . . . substitute for x

3y² = 36y +47 . . . . . . . eliminate parentheses

3y² -36y -47 = 0 . . . . subtract the right-side expression

b) Solutions

The equation can be written in vertex form as ...

3y² -36y = 47

3(y² -12y +36) = 47 +3(36)

3(y -6)² = 155

y -6 = ±√(155/3)

y = 6 ± (√465)/3 ≈ -1.19, 13.19

x = 9y +10 ≈ -0.69, 128.69

The approximate solutions are ...

(x, y) ≈ (-0.7, -1.2), (128.7, 13.2)

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Additional comment

The attached graph confirms these solution points.

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Answer 2

see explanation

Explanation:

(a)

x - 9y = 10 ( add 9y to both sides )

x = 9y + 10 → (1)

3y² = 4x + 7 → (2)

substitute x = 9y + 10 into (2)

3y² = 4(9y + 10) + 7

3y² = 36y + 40 + 7

3y² = 36y + 47 ( subtract 36y + 47 from both sides )

3y² - 36y - 47 = 0 ← as required

(b)

solving using the method of completing the square

add 47 to both sides

3y² - 36y = 47 ← factor out 3 from each term in the left side

3(y² - 12y) = 47

to complete the square on the left

add/subtract ( half the coefficient of the y- term)² to y² - 12y

3(y² + 2(- 6)y + 36 - 36) = 47

3(y - 6)² - 3(36) = 47

3(y - 6)² - 108 = 47 ( add 108 to both sides )

3(y - 6)² = 155 ( divide both sides by 3 )

(y - 6)² =
`(155)/(3)` ( take square root of both sides )

y - 6 = ±
`\sqrt{(155)/(3) }` ≈ ± 7.2 ( to 1 decimal place )

add 6 to both sides

y = 6 ± 7.2

then

y = 6 - 7.2 = - 1.2

y = 6 + 7.2 = 13.2

substitute these values into ( 1) for corresponding values of x

x = - 1.2 : x = 9(- 1.2) + 10 = - 10.8 + 10 = - 0.8 ⇒ (- 1.2, - 0.8 )

x = 13.2 : x = 9(13.2) + 10 = 118.8 + 10 = 128.8 ⇒ (13.2, 128.8 )