2 5 rational equations word-problems | PPT - Free Printable

Problem Statement:

We are given a word problem involving rational equations. The task is to determine the speed \( x \) (in mph) at which we traveled from point A to point B. Here's the breakdown of the problem:

- Distance from A to B: 6 miles.
- Speed from A to B: \( x \) mph.
- Speed on the return trip from B to A: \( x + 1 \) mph (1 mph faster).
- Total time for the round trip: 5 hours.

We need to solve for \( x \).

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Step-by-Step Solution:

#### 1. Set up the table:
We use the formula \( \text{Time} = \frac{\text{Distance}}{\text{Rate}} \) to organize the information.

| Direction | Distance (miles) | Rate (mph) | Time (hours) |
|-----------|------------------|------------|--------------|
| Go | 6 | \( x \) | \( \frac{6}{x} \) |
| Return | 6 | \( x + 1 \) | \( \frac{6}{x + 1} \) |

#### 2. Express the total time:
The total time for the round trip is the sum of the time taken to go from A to B and the time taken to return from B to A. According to the problem, this total time is 5 hours. Therefore, we can write the equation:

\[
\frac{6}{x} + \frac{6}{x + 1} = 5
\]

#### 3. Combine the fractions:
To combine the fractions, we need a common denominator. The common denominator for \( x \) and \( x + 1 \) is \( x(x + 1) \). Rewrite each fraction with this common denominator:

\[
\frac{6}{x} = \frac{6(x + 1)}{x(x + 1)}
\]
\[
\frac{6}{x + 1} = \frac{6x}{x(x + 1)}
\]

Now, add the fractions:

\[
\frac{6(x + 1)}{x(x + 1)} + \frac{6x}{x(x + 1)} = 5
\]

Combine the numerators:

\[
\frac{6(x + 1) + 6x}{x(x + 1)} = 5
\]

Simplify the numerator:

\[
6(x + 1) + 6x = 6x + 6 + 6x = 12x + 6
\]

So the equation becomes:

\[
\frac{12x + 6}{x(x + 1)} = 5
\]

#### 4. Eliminate the denominator:
To eliminate the denominator, multiply both sides of the equation by \( x(x + 1) \):

\[
12x + 6 = 5x(x + 1)
\]

Expand the right-hand side:

\[
12x + 6 = 5x^2 + 5x
\]

#### 5. Rearrange into standard quadratic form:
Move all terms to one side of the equation to set it equal to zero:

\[
5x^2 + 5x - 12x - 6 = 0
\]

Simplify:

\[
5x^2 - 7x - 6 = 0
\]

#### 6. Solve the quadratic equation:
We solve the quadratic equation \( 5x^2 - 7x - 6 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 5 \), \( b = -7 \), and \( c = -6 \). Substitute these values into the formula:

\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(5)(-6)}}{2(5)}
\]

Simplify step by step:

\[
x = \frac{7 \pm \sqrt{49 + 120}}{10}
\]
\[
x = \frac{7 \pm \sqrt{169}}{10}
\]
\[
x = \frac{7 \pm 13}{10}
\]

This gives us two potential solutions:

\[
x = \frac{7 + 13}{10} = \frac{20}{10} = 2
\]
\[
x = \frac{7 - 13}{10} = \frac{-6}{10} = -0.6
\]

Since speed cannot be negative, we discard \( x = -0.6 \). Thus, the valid solution is:

\[
x = 2
\]

#### 7. Verify the solution:
If \( x = 2 \):
- Speed from A to B: \( x = 2 \) mph.
- Speed from B to A: \( x + 1 = 3 \) mph.
- Time from A to B: \( \frac{6}{2} = 3 \) hours.
- Time from B to A: \( \frac{6}{3} = 2 \) hours.
- Total time: \( 3 + 2 = 5 \) hours, which matches the given total time.

The solution is correct.

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Final Answer:

\[
\boxed{2}
\]