Question

Using the properties of equality, find the solution to the equation 2/f/5−8/5+2f=8.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the solution to the equation 2/f/5 − 8/5 + 2f = 8, we can simplify the equation using the properties of equality.

First, let's determine the LCD (Least Common Denominator) for the fractions in the equation. In this case, the LCD is 5f because the terms involve both f and 5.

Now, multiply each term by the LCD to clear the fractions:

2/f/5 * 5f - 8/5 * 5f + 2f * 5f = 8 * 5f

The f's in 2/f/5 * 5f cancel out, leaving us with 2 * 5 = 10:

10 - 8 * 5f + 2f * 5f = 40f

Expanding the terms:

10 - 40f + 10f^2 = 40f

Rearranging the terms:

10f^2 - 40f + 40f - 10 = 0

Simplifying:

10f^2 - 10 = 0

Now divide both sides of the equation by 10:

f^2 - 1 = 0

Rearranging the terms:

f^2 = 1

Taking the square root of both sides:

f = ±√1

Therefore, the solutions to the equation 2/f/5 − 8/5 + 2f = 8 are f = 1 and f = -1.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To solve the equation 2/f/5 - 8/5 + 2f = 8, we can follow these steps:

Step 1: Simplify the expression on the left side.

The expression 2/f/5 can be written as 2/(f/5), which is equivalent to 2 * (5/f) or 10/f.

So, the equation becomes 10/f - 8/5 + 2f = 8.

Step 2: Find a common denominator.

To combine the fractions, we need to find a common denominator. The common denominator for f and 5 is 5f.

Step 3: Rewrite the fractions with the common denominator.

10/f can be rewritten as (10 * 5) / (f * 5) or 50/5f.

The equation now becomes 50/5f - 8/5 + 2f = 8.

Step 4: Combine the fractions.

Now that we have the fractions with the common denominator, we can combine them. The equation becomes (50 - 8f + 10f) / 5f = 8.

Step 5: Simplify and rearrange the equation.

Combine the terms 50 - 8f + 10f as 2f + 50.

The equation becomes (2f + 50) / 5f = 8.

Cross-multiply to eliminate the fraction: 2f + 50 = 8 * 5f.

Simplify further: 2f + 50 = 40f.

Step 6: Solve for f.

To solve for f, we need to isolate the variable. Let's start by getting all the terms with f on one side of the equation.

Subtract 2f from both sides: 50 = 40f - 2f.

Combine like terms: 50 = 38f.

Divide both sides by 38: f = 50/38.

Simplify the fraction: f ≈ 1.32.

Therefore, the solution to the equation 2/f/5 - 8/5 + 2f = 8 is approximately f ≈ 1.32.

Answer 3

[AI-generated answer — Explain Bot]

To find the solution to the equation, we will use the properties of equality to isolate the variable f.

1. Start by simplifying both sides of the equation. Combine the fractions 2/f/5 and 8/5 by finding a common denominator, which is 5f.

(2/f/5) - (8/5) + 2f = 8
(2 * 5 - f * 8 + 2f * 5f) / (f * 5) = 8
(10 - 8f + 10f^2) / (5f) = 8

2. Next, multiply both sides of the equation by 5f to eliminate the denominator on the left side.

10 - 8f + 10f^2 = 40f

3. Rearrange the equation to have all terms on one side.

10f^2 - 48f + 10 = 0

4. Now, we have a quadratic equation. To solve for f, we can factor, complete the square, or use the quadratic formula. In this case, let's use the quadratic formula.

The quadratic formula is given by:
f = (-b ± √(b^2 - 4ac)) / (2a)

For our equation 10f^2 - 48f + 10 = 0, the coefficients are:
a = 10
b = -48
c = 10

Plugging these values into the quadratic formula, we get:

f = (-(-48) ± √((-48)^2 - 4 * 10 * 10)) / (2 * 10)
f = (48 ± √(2304 - 400)) / 20
f = (48 ± √(1904)) / 20
f = (48 ± √(4 * 476)) / 20
f = (48 ± 2√(119)) / 20
f = (24 ± √(119)) / 10

This gives us two possible solutions for f:

f = (24 + √(119)) / 10
f = (24 - √(119)) / 10

Therefore, the solutions to the equation 2/f/5 - 8/5 + 2f = 8 are (24 + √(119)) / 10 and (24 - √(119)) / 10.