https://file.aiquickdraw.com/mathbot/file/172702604499166ck5ymd.jpg

Let's solve each of these equations step by step:

### Problem 7:
\[
4(3x - 10) + 7 = 15
\]
**Step 1:** Expand the equation.
\[
4(3x - 10) = 12x - 40
\]
So the equation becomes:
\[
12x - 40 + 7 = 15
\]

**Step 2:** Combine like terms.
\[
12x - 33 = 15
\]

**Step 3:** Add 33 to both sides.
\[
12x = 48
\]

**Step 4:** Divide both sides by 12.
\[
x = 4
\]

### Problem 8:
\[
6 - 5(2x + 1) = 21
\]
**Step 1:** Expand the equation.
\[
6 - 5(2x + 1) = 6 - 10x - 5
\]
Simplify it:
\[
1 - 10x = 21
\]

**Step 2:** Subtract 1 from both sides.
\[
-10x = 20
\]

**Step 3:** Divide both sides by -10.
\[
x = -2
\]

### Problem 9:
\[
2 - \frac{3}{4}(8x - 6) = 11
\]
**Step 1:** Distribute \(\frac{3}{4}\) across the terms inside the parentheses.
\[
\frac{3}{4}(8x - 6) = \frac{3}{4} \times 8x - \frac{3}{4} \times 6 = 6x - 4.5
\]
So the equation becomes:
\[
2 - (6x - 4.5) = 11
\]
This simplifies to:
\[
2 - 6x + 4.5 = 11
\]

**Step 2:** Combine like terms.
\[
6.5 - 6x = 11
\]

**Step 3:** Subtract 6.5 from both sides.
\[
-6x = 4.5
\]

**Step 4:** Divide both sides by -6.
\[
x = -\frac{3}{4} \quad \text{or} \quad x = -0.75
\]

### Problem 10:
\[
0.2(4 - 5x) + 1 = 2.4
\]
**Step 1:** Expand the equation.
\[
0.2(4 - 5x) = 0.8 - x
\]
So the equation becomes:
\[
0.8 - x + 1 = 2.4
\]

**Step 2:** Combine like terms.
\[
1.8 - x = 2.4
\]

**Step 3:** Subtract 1.8 from both sides.
\[
-x = 0.6
\]

**Step 4:** Multiply both sides by -1.
\[
x = -0.6
\]

---

If you'd like any further details or have questions about specific steps, feel free to ask!

### Related Questions:
1. How do you distribute terms across parentheses in equations?
2. How do you handle fractions in equations, like in problem 9?
3. What are the steps for isolating the variable \(x\) in an equation?
4. How can you check if your solution for \(x\) is correct?
5. What are like terms, and why do we combine them in equations?

### Tip:
Always simplify your equation by combining like terms and eliminating fractions or decimals early in the process.

Solve each equation. Justify your solution steps, and check each solution. 1) 4(3x - 10) + 7 = 15, 2) 6 - 5(2x + 1) = 21, 3) 2 - (3/4)(8x - 6) = 11, 4) 0.2(4 - 5x) + 1 = 2.4

This guide provides a step-by-step solution for solving linear equations like 4(3x - 10) + 7 = 15 and 0.2(4 - 5x) + 1 = 2.4. Learn to apply the distributive property, combine like terms, and solve for x. Suitable for students in Grades 8-10.

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