Answer :
Certainly! Let's go through each of the equations step-by-step and verify their solutions.
### 1. Solve for [tex]\( x \)[/tex]:
[tex]\[ 3x = 2x + 18 \][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ 3x - 2x = 18 \][/tex]
[tex]\[ x = 18 \][/tex]
Solution: [tex]\( x = 18 \)[/tex]
### 4. Solve for [tex]\( z \)[/tex]:
[tex]\[ 4z + 3 = 6 + 2z \][/tex]
Subtract [tex]\( 2z \)[/tex] from both sides:
[tex]\[ 4z - 2z + 3 = 6 \][/tex]
[tex]\[ 2z + 3 = 6 \][/tex]
Subtract 3 from both sides:
[tex]\[ 2z = 3 \][/tex]
Divide both sides by 2:
[tex]\[ z = \frac{3}{2} \][/tex]
Solution: [tex]\( z = \frac{3}{2} \)[/tex]
### 2. Solve for [tex]\( t \)[/tex]:
[tex]\[ 5t - 3 = 3t - 5 \][/tex]
Subtract [tex]\( 3t \)[/tex] from both sides:
[tex]\[ 5t - 3t - 3 = -5 \][/tex]
[tex]\[ 2t - 3 = -5 \][/tex]
Add 3 to both sides:
[tex]\[ 2t = -2 \][/tex]
Divide both sides by 2:
[tex]\[ t = -1 \][/tex]
Solution: [tex]\( t = -1 \)[/tex]
### 5. Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - 1 = 14 - x \][/tex]
Add [tex]\( x \)[/tex] to both sides:
[tex]\[ 2x + x - 1 = 14 \][/tex]
[tex]\[ 3x - 1 = 14 \][/tex]
Add 1 to both sides:
[tex]\[ 3x = 15 \][/tex]
Divide both sides by 3:
[tex]\[ x = 5 \][/tex]
Solution: [tex]\( x = 5 \)[/tex]
### 3. Solve for [tex]\( x \)[/tex]:
[tex]\[ 5x + 9 = 5 + 3x \][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 5x - 3x + 9 = 5 \][/tex]
[tex]\[ 2x + 9 = 5 \][/tex]
Subtract 9 from both sides:
[tex]\[ 2x = -4 \][/tex]
Divide both sides by 2:
[tex]\[ x = -2 \][/tex]
Solution: [tex]\( x = -2 \)[/tex]
### 6. Solve for [tex]\( x \)[/tex]:
[tex]\[ 8x + 4 = 3(x - 1) + 7 \][/tex]
Distribute the 3 on the right side:
[tex]\[ 8x + 4 = 3x - 3 + 7 \][/tex]
[tex]\[ 8x + 4 = 3x + 4 \][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 8x - 3x + 4 = 4 \][/tex]
[tex]\[ 5x + 4 = 4 \][/tex]
Subtract 4 from both sides:
[tex]\[ 5x = 0 \][/tex]
Divide both sides by 5:
[tex]\[ x = 0 \][/tex]
Solution: [tex]\( x = 0 \)[/tex]
### 7. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{4}{5}(x + 10) \][/tex]
Multiply both sides by 5 to eliminate the fraction:
[tex]\[ 5x = 4(x + 10) \][/tex]
Distribute the 4 on the right side:
[tex]\[ 5x = 4x + 40 \][/tex]
Subtract [tex]\( 4x \)[/tex] from both sides:
[tex]\[ 5x - 4x = 40 \][/tex]
[tex]\[ x = 40 \][/tex]
Solution: [tex]\( x = 40 \)[/tex]
### 8. Solve for [tex]\( x \)[/tex]:
[tex]\[ \frac{2x}{3} + 1 = \frac{7x}{15} + 3 \][/tex]
To simplify, find a common denominator for the fractions involved, which is 15:
[tex]\[ \frac{10x}{15} + 1 = \frac{7x}{15} + 3 \][/tex]
Multiply through by 15 to clear denominators:
[tex]\[ 10x + 15 = 7x + 45 \][/tex]
Subtract [tex]\( 7x \)[/tex] from both sides:
[tex]\[ 10x - 7x + 15 = 45 \][/tex]
[tex]\[ 3x + 15 = 45 \][/tex]
Subtract 15 from both sides:
[tex]\[ 3x = 30 \][/tex]
Divide both sides by 3:
[tex]\[ x = 10 \][/tex]
Solution: [tex]\( x = 10 \)[/tex]
### 10. Solve for [tex]\( m \)[/tex]:
[tex]\[ 3m = 5m - \frac{8}{5} \][/tex]
Subtract [tex]\( 5m \)[/tex] from both sides:
[tex]\[ 3m - 5m = -\frac{8}{5} \][/tex]
[tex]\[ -2m = -\frac{8}{5} \][/tex]
Divide both sides by -2:
[tex]\[ m = \frac{8}{5 \cdot 2} \][/tex]
[tex]\[ m = \frac{8}{10} \][/tex]
[tex]\[ m = 0.8 \][/tex]
Solution: [tex]\( m = 0.8 \)[/tex]
### 9. Solve for [tex]\( y \)[/tex]:
[tex]\[ 2y + \frac{5}{3} = \frac{26}{3} - y \][/tex]
Multiply through by 3 to clear denominators:
[tex]\[ 6y + 5 = 26 - 3y \][/tex]
Add [tex]\( 3y \)[/tex] to both sides:
[tex]\[ 6y + 3y + 5 = 26 \][/tex]
[tex]\[ 9y + 5 = 26 \][/tex]
Subtract 5 from both sides:
[tex]\[ 9y = 21 \][/tex]
Divide both sides by 9:
[tex]\[ y = \frac{21}{9} \][/tex]
[tex]\[ y = \frac{7}{3} \approx 2.333 \][/tex]
Solution: [tex]\( y = \frac{7}{3} \approx 2.333 \)[/tex]
In summary:
1. [tex]\( x = 18 \)[/tex]
2. [tex]\( t = -1 \)[/tex]
3. [tex]\( x = -2 \)[/tex]
4. [tex]\( z = \frac{3}{2} \)[/tex]
5. [tex]\( x = 5 \)[/tex]
6. [tex]\( x = 0 \)[/tex]
7. [tex]\( x = 40 \)[/tex]
8. [tex]\( x = 10 \)[/tex]
9. [tex]\( y = \frac{7}{3} \approx 2.333 \)[/tex]
10. [tex]\( m = 0.8 \)[/tex]
### 1. Solve for [tex]\( x \)[/tex]:
[tex]\[ 3x = 2x + 18 \][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ 3x - 2x = 18 \][/tex]
[tex]\[ x = 18 \][/tex]
Solution: [tex]\( x = 18 \)[/tex]
### 4. Solve for [tex]\( z \)[/tex]:
[tex]\[ 4z + 3 = 6 + 2z \][/tex]
Subtract [tex]\( 2z \)[/tex] from both sides:
[tex]\[ 4z - 2z + 3 = 6 \][/tex]
[tex]\[ 2z + 3 = 6 \][/tex]
Subtract 3 from both sides:
[tex]\[ 2z = 3 \][/tex]
Divide both sides by 2:
[tex]\[ z = \frac{3}{2} \][/tex]
Solution: [tex]\( z = \frac{3}{2} \)[/tex]
### 2. Solve for [tex]\( t \)[/tex]:
[tex]\[ 5t - 3 = 3t - 5 \][/tex]
Subtract [tex]\( 3t \)[/tex] from both sides:
[tex]\[ 5t - 3t - 3 = -5 \][/tex]
[tex]\[ 2t - 3 = -5 \][/tex]
Add 3 to both sides:
[tex]\[ 2t = -2 \][/tex]
Divide both sides by 2:
[tex]\[ t = -1 \][/tex]
Solution: [tex]\( t = -1 \)[/tex]
### 5. Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - 1 = 14 - x \][/tex]
Add [tex]\( x \)[/tex] to both sides:
[tex]\[ 2x + x - 1 = 14 \][/tex]
[tex]\[ 3x - 1 = 14 \][/tex]
Add 1 to both sides:
[tex]\[ 3x = 15 \][/tex]
Divide both sides by 3:
[tex]\[ x = 5 \][/tex]
Solution: [tex]\( x = 5 \)[/tex]
### 3. Solve for [tex]\( x \)[/tex]:
[tex]\[ 5x + 9 = 5 + 3x \][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 5x - 3x + 9 = 5 \][/tex]
[tex]\[ 2x + 9 = 5 \][/tex]
Subtract 9 from both sides:
[tex]\[ 2x = -4 \][/tex]
Divide both sides by 2:
[tex]\[ x = -2 \][/tex]
Solution: [tex]\( x = -2 \)[/tex]
### 6. Solve for [tex]\( x \)[/tex]:
[tex]\[ 8x + 4 = 3(x - 1) + 7 \][/tex]
Distribute the 3 on the right side:
[tex]\[ 8x + 4 = 3x - 3 + 7 \][/tex]
[tex]\[ 8x + 4 = 3x + 4 \][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 8x - 3x + 4 = 4 \][/tex]
[tex]\[ 5x + 4 = 4 \][/tex]
Subtract 4 from both sides:
[tex]\[ 5x = 0 \][/tex]
Divide both sides by 5:
[tex]\[ x = 0 \][/tex]
Solution: [tex]\( x = 0 \)[/tex]
### 7. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{4}{5}(x + 10) \][/tex]
Multiply both sides by 5 to eliminate the fraction:
[tex]\[ 5x = 4(x + 10) \][/tex]
Distribute the 4 on the right side:
[tex]\[ 5x = 4x + 40 \][/tex]
Subtract [tex]\( 4x \)[/tex] from both sides:
[tex]\[ 5x - 4x = 40 \][/tex]
[tex]\[ x = 40 \][/tex]
Solution: [tex]\( x = 40 \)[/tex]
### 8. Solve for [tex]\( x \)[/tex]:
[tex]\[ \frac{2x}{3} + 1 = \frac{7x}{15} + 3 \][/tex]
To simplify, find a common denominator for the fractions involved, which is 15:
[tex]\[ \frac{10x}{15} + 1 = \frac{7x}{15} + 3 \][/tex]
Multiply through by 15 to clear denominators:
[tex]\[ 10x + 15 = 7x + 45 \][/tex]
Subtract [tex]\( 7x \)[/tex] from both sides:
[tex]\[ 10x - 7x + 15 = 45 \][/tex]
[tex]\[ 3x + 15 = 45 \][/tex]
Subtract 15 from both sides:
[tex]\[ 3x = 30 \][/tex]
Divide both sides by 3:
[tex]\[ x = 10 \][/tex]
Solution: [tex]\( x = 10 \)[/tex]
### 10. Solve for [tex]\( m \)[/tex]:
[tex]\[ 3m = 5m - \frac{8}{5} \][/tex]
Subtract [tex]\( 5m \)[/tex] from both sides:
[tex]\[ 3m - 5m = -\frac{8}{5} \][/tex]
[tex]\[ -2m = -\frac{8}{5} \][/tex]
Divide both sides by -2:
[tex]\[ m = \frac{8}{5 \cdot 2} \][/tex]
[tex]\[ m = \frac{8}{10} \][/tex]
[tex]\[ m = 0.8 \][/tex]
Solution: [tex]\( m = 0.8 \)[/tex]
### 9. Solve for [tex]\( y \)[/tex]:
[tex]\[ 2y + \frac{5}{3} = \frac{26}{3} - y \][/tex]
Multiply through by 3 to clear denominators:
[tex]\[ 6y + 5 = 26 - 3y \][/tex]
Add [tex]\( 3y \)[/tex] to both sides:
[tex]\[ 6y + 3y + 5 = 26 \][/tex]
[tex]\[ 9y + 5 = 26 \][/tex]
Subtract 5 from both sides:
[tex]\[ 9y = 21 \][/tex]
Divide both sides by 9:
[tex]\[ y = \frac{21}{9} \][/tex]
[tex]\[ y = \frac{7}{3} \approx 2.333 \][/tex]
Solution: [tex]\( y = \frac{7}{3} \approx 2.333 \)[/tex]
In summary:
1. [tex]\( x = 18 \)[/tex]
2. [tex]\( t = -1 \)[/tex]
3. [tex]\( x = -2 \)[/tex]
4. [tex]\( z = \frac{3}{2} \)[/tex]
5. [tex]\( x = 5 \)[/tex]
6. [tex]\( x = 0 \)[/tex]
7. [tex]\( x = 40 \)[/tex]
8. [tex]\( x = 10 \)[/tex]
9. [tex]\( y = \frac{7}{3} \approx 2.333 \)[/tex]
10. [tex]\( m = 0.8 \)[/tex]
