Individual Practice:

1. Solve 2  =   3
              ----      -----
             x + 7  x + 4

2. Solve r  +  r - 1 = 7
              ---     ------    ----
              3        r        6

Let's go through each equation step by step:

### 1. Solve \( \frac{2}{x + 7} = \frac{3}{x + 4} \)

We want to solve for \(x\). To eliminate the denominators, we can cross-multiply:

\[
2(x + 4) = 3(x + 7)
\]

Expand both sides:

\[
2x + 8 = 3x + 21
\]

Now, isolate \(x\) by first subtracting \(2x\) from both sides:

\[
8 = x + 21
\]

Then subtract 21 from both sides:

\[
8 - 21 = x
\]

\[
x = -13
\]

### 2. Solve \( \frac{r}{3} + \frac{r - 1}{r} = \frac{7}{6} \)

To eliminate the fractions, find the least common denominator (LCD). The LCD here is \(6r\). Multiply every term by \(6r\) to eliminate the fractions:

\[
6r \times \frac{r}{3} + 6r \times \frac{r - 1}{r} = 6r \times \frac{7}{6}
\]

Simplifying each term:

\[
2r^2 + 6(r - 1) = 7r
\]

Expand the expression:

\[
2r^2 + 6r - 6 = 7r
\]

Now, subtract \(7r\) from both sides to get a quadratic equation:

\[
2r^2 + 6r - 6 - 7r = 0
\]

\[
2r^2 - r - 6 = 0
\]

This is a quadratic equation. Use the quadratic formula to solve for \(r\):

\[
r = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-6)}}{2(2)}
\]

Simplifying the discriminant:

\[
r = \frac{1 \pm \sqrt{1 + 48}}{4}
\]

\[
r = \frac{1 \pm \sqrt{49}}{4}
\]

\[
r = \frac{1 \pm 7}{4}
\]

Now solve for both possible values of \(r\):

1. \(r = \frac{1 + 7}{4} = \frac{8}{4} = 2\)
2. \(r = \frac{1 - 7}{4} = \frac{-6}{4} = -\frac{3}{2}\)

Thus, the solutions for \(r\) are \(r = 2\) and \(r = -\frac{3}{2}\).

---

Let me know if you'd like any further clarifications!

Here are 5 related questions for further practice:
1. Solve \( \frac{4}{x + 3} = \frac{5}{x + 6} \).
2. Solve \( \frac{r}{4} + \frac{r + 2}{r} = \frac{5}{6} \).
3. Solve \( \frac{5}{x - 1} = \frac{3}{x + 2} \).
4. Solve \( \frac{2r}{5} + \frac{r - 2}{r} = \frac{3}{4} \).
5. Solve \( \frac{x + 2}{x + 5} = \frac{x - 1}{x + 7} \).

**Tip:** When solving equations with fractions, cross-multiplying can quickly eliminate denominators. Just be careful to expand and simplify correctly!

Individual Practice:

1. Solve 2 / (x + 7) = 3 / (x + 4)

2. Solve r / 3 + (r - 1) / r = 7 / 6

Learn how to solve rational equations like 2/(x + 7) = 3/(x + 4) and quadratic equations formed from expressions like r/3 + (r - 1)/r = 7/6. This solution includes cross multiplication and the quadratic formula, making it suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation