Let's solve each equation step-by-step:
### 1. [tex]\(3(3x + 1) - 10(x + 1) = 0\)[/tex]
1. Distribute the multiplication:
[tex]\[
9x + 3 - 10x - 10 = 0
\][/tex]
2. Combine like terms:
[tex]\[
(9x - 10x) + (3 - 10) = 0
\][/tex]
[tex]\[
-x - 7 = 0
\][/tex]
3. Isolate [tex]\(x\)[/tex]:
[tex]\[
-x = 7
\][/tex]
[tex]\[
x = -7
\][/tex]
So, the solution is [tex]\(x = -7\)[/tex].
### 2. [tex]\(5(x + 2) + 3(x - 5) = 19\)[/tex]
1. Distribute the multiplication:
[tex]\[
5x + 10 + 3x - 15 = 19
\][/tex]
2. Combine like terms:
[tex]\[
(5x + 3x) + (10 - 15) = 19
\][/tex]
[tex]\[
8x - 5 = 19
\][/tex]
3. Isolate [tex]\(x\)[/tex]:
[tex]\[
8x = 24
\][/tex]
[tex]\[
x = 3
\][/tex]
So, the solution is [tex]\(x = 3\)[/tex].
### 3. [tex]\(4x + 3x - 7 = 5x - (35 - 6x)\)[/tex]
1. Simplify both sides:
[tex]\[
7x - 7 = 5x - 35 + 6x
\][/tex]
2. Distribute and combine like terms:
[tex]\[
7x - 7 = 11x - 35
\][/tex]
3. Isolate [tex]\(x\)[/tex]:
[tex]\[
7x - 11x = -35 + 7
\][/tex]
[tex]\[
-4x = -28
\][/tex]
[tex]\[
x = 7
\][/tex]
So, the solution is [tex]\(x = 7\)[/tex].
### 4. [tex]\(4 - 2 - 2(x - 9) - 6(x - 5) = 50 - 6x - 21\)[/tex]
1. Simplify both sides:
[tex]\[
2 - 2(x - 9) - 6(x - 5) = 50 - 21 - 6x
\][/tex]
[tex]\[
2 - 2x + 18 - 6x + 30 = 29 - 6x
\][/tex]
2. Distribute and combine like terms:
[tex]\[
50 - 8x = 29 - 6x
\][/tex]
3. Isolate [tex]\(x\)[/tex]:
[tex]\[
-8x + 6x = 29 - 50
\][/tex]
[tex]\[
-2x = -21
\][/tex]
[tex]\[
x = \frac{21}{2} \quad \text{or} \quad x = 10.5
\][/tex]
So, the solution is [tex]\(x = \frac{21}{2}\)[/tex] or [tex]\(x = 10.5\)[/tex].
### 5. [tex]\(5(x + 3) = 2(3x - 1) + 3(2x + 1)\)[/tex]
1. Distribute the multiplication:
[tex]\[
5x + 15 = 6x - 2 + 6x + 3
\][/tex]
2. Combine like terms and simplify:
[tex]\[
5x + 15 = 12x + 1
\][/tex]
3. Isolate [tex]\(x\)[/tex]:
[tex]\[
5x - 12x = 1 - 15
\][/tex]
[tex]\[
-7x = -14
\][/tex]
[tex]\[
x = 2
\][/tex]
So, the solution is [tex]\(x = 2\)[/tex].
### Summary
The solutions are:
1. [tex]\(x = -7\)[/tex]
2. [tex]\(x = 3\)[/tex]
3. [tex]\(x = 7\)[/tex]
4. [tex]\(x = \frac{21}{2}\)[/tex]
5. [tex]\(x = 2\)[/tex]
