https://file.aiquickdraw.com/mathbot/file/172810510958672cidcm8.jpg

Let's work through the equations you uploaded:

### Solve the following equations:

**a.** \( 19 + x = -12 \)

Subtract 19 from both sides:
\[
x = -12 - 19 = -31
\]
**Solution:** \( x = -31 \)

---

**b.** \( 15 - 4x = 11 \)

Subtract 15 from both sides:
\[
-4x = 11 - 15 = -4
\]
Divide both sides by -4:
\[
x = \frac{-4}{-4} = 1
\]
**Solution:** \( x = 1 \)

---

**c.** \( \frac{3}{2x} = -9 \)

Multiply both sides by \( 2x \) to eliminate the denominator:
\[
3 = -18x
\]
Divide both sides by -18:
\[
x = \frac{3}{-18} = -\frac{1}{6}
\]
**Solution:** \( x = -\frac{1}{6} \)

---

**d.** \( 3x - 4x = 11 \)

Simplify:
\[
-x = 11
\]
Multiply both sides by -1:
\[
x = -11
\]
**Solution:** \( x = -11 \)

---

**e.** \( \frac{3x}{7} = \frac{4}{3} \)

Cross-multiply:
\[
3x \times 3 = 4 \times 7
\]
\[
9x = 28
\]
Divide both sides by 9:
\[
x = \frac{28}{9}
\]
**Solution:** \( x = \frac{28}{9} \)

---

**f.** \( \frac{3x + 5}{2} = 8 \)

Multiply both sides by 2:
\[
3x + 5 = 16
\]
Subtract 5 from both sides:
\[
3x = 11
\]
Divide both sides by 3:
\[
x = \frac{11}{3}
\]
**Solution:** \( x = \frac{11}{3} \)

---

**g.** \( \frac{4x + 7}{7} = 9 \)

Multiply both sides by 7:
\[
4x + 7 = 63
\]
Subtract 7 from both sides:
\[
4x = 56
\]
Divide both sides by 4:
\[
x = 14
\]
**Solution:** \( x = 14 \)

---

**i.** \( \frac{y}{3} + 2 = 2 \)

Subtract 2 from both sides:
\[
\frac{y}{3} = 0
\]
Multiply both sides by 3:
\[
y = 0
\]
**Solution:** \( y = 0 \)

---

**j.** \( \frac{3x + 1}{3} = \frac{5x - 2}{8} \)

Cross-multiply:
\[
8(3x + 1) = 3(5x - 2)
\]
Expand both sides:
\[
24x + 8 = 15x - 6
\]
Move all terms involving \(x\) to one side and constants to the other side:
\[
24x - 15x = -6 - 8
\]
\[
9x = -14
\]
Divide both sides by 9:
\[
x = \frac{-14}{9}
\]
**Solution:** \( x = \frac{-14}{9} \)

---

Would you like further details or have any questions on any step? 

Here are 5 related questions to explore further:
1. How would you check if your solution for \( x = \frac{28}{9} \) in equation **e** is correct?
2. What happens if you have a quadratic equation? How would you solve it?
3. Can you explain how cross-multiplying works in solving rational equations?
4. Why is it necessary to isolate the variable when solving an equation?
5. How does simplifying an equation first (like in **d**) make the solution easier?

**Tip:** When working with fractions, cross-multiplication is a powerful tool for solving equations quickly!

Solve the following equations as shown in the image.

Step-by-step guide on how to solve linear and rational equations involving fractions and cross-multiplication. Topics covered include isolating variables, cross-multiplication, and solving for unknowns, suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation