Match the equations of ellipses to their equivalent equations in standard form.

# The general form of  x²/3² + (y - 2)²/2² = 1 is 4x² - 36y + 9y² = 0

# The general form of (x + 7)²/7² + y²/6² = 1 is 36x² + 504x + 49y² = 0

# The general form of x²/6² + (y + 4)²/4² = 1  is 16x² + 288y + 36y² = 0

# The general form of (x - 3)²/3² + y²/5² = 1  is 25x² - 150x + 9y² = 0

Step-by-step explanation:

* Lets revise the general form and the standard form of the ellipse

- The general form is Ax² + Bxy + Cy² + Dx + Ey + F = 0

- The standard form is (x - h)²/a² + (y - k)²/b² = 1

* Lets solve the problem

# x²/3² + (y - 2)²/2² = 1

∵ x²/9 + (y - 2)²/4 = 1 ⇒ multiply both sides by 9 × 4 (36)

∴ 4x² + 9(y - 2)² = 36 ⇒ open the bracket power 2

∴ 4x² + 9(y² - 4y + 4) = 36 ⇒ multiply the bracket by 9

∴ 4x² + 9y² - 36y + 36 = 36 ⇒ subtract 36 from both sides

∴ 4x² - 36y + 9y² = 0

* The general form of  x²/3² + (y - 2)²/2² = 1 is 4x² + 9y² - 36y = 0

# (x + 7)²/7² + y²/6² = 1

∵ (x + 7)²/49 + y²/36 = 1 ⇒ multiply both sides by 49 × 36 (1764)

∴ 36(x + 7)² + 49y² = 1764 ⇒ open the bracket power 2

∴ 36(x² + 14x + 49) + 49y² = 1764 ⇒ multiply the bracket by 36

∴ 36x² + 504x + 1764 + 49y² = 1764 ⇒ subtract 1764 from both sides

∴ 36x² + 504x + 49y² = 0

* The general form of (x + 7)²/7² + y²/6² = 1 is 36x² + 504x + 49y² = 0

# x²/6² + (y + 4)²/4² = 1

∵ x²/36 + (y + 4)²/16 = 1 ⇒ multiply both sides by 36 × 16 (576)

∴ 16x² + 36(y + 4)² = 576 ⇒ open the bracket power 2

∴ 16x² + 36(y² + 8y + 16) = 576 ⇒ multiply the bracket by 36

∴ 16x² + 36y² + 288y + 576 = 576 ⇒ subtract 576 from both sides

∴ 16x² + 288y + 36y² = 0

* The general form of x²/6² + (y + 4)²/4² = 1  is 16x² + 288y + 36y² = 0

# (x - 3)²/3² + y²/5² = 1

∵ (x - 3)²/9 + y²/25 = 1 ⇒ multiply both sides by 9 × 25 (225)

∴ 25(x - 3)² + 9y² = 225 ⇒ open the bracket power 2

∴ 25(x² - 6x + 9) + 9y² = 225 ⇒ multiply the bracket by 25

∴ 25x² - 150x + 225 + 9y² = 225 ⇒ subtract 225 from both sides

∴ 25x² - 150x + 9y² = 0

* The general form of (x - 3)²/3² + y²/5² = 1  is 25x² - 150x + 9y² = 0

the final amount you will pay is $68.04.

the tip is determined by the pre-tax amount, so we calculate the tip based on the original bill, not after we've added sales tax.

to find the tip:

we can divide 54.00 by 100 to get 1% of 54. then we multiply that by 18.

54 / 100 = .54

.54 x 18 = 9.72

tip = 9.72$

to find the tax:

we can divide 54.00 by 100 to get 1% of 54. then we can multiply that by 8 to get 8%.

54 / 100 = .54

.54 x 8 = 4.32

tax= 4.32

now we can add the tip and tax to the bill to get our total.

54 + 9.72 + 4.32 = 68.04

total: $68.04

step-by-step explanation: