Answer :
Let's solve each equation one by one and verify the results:
1. [tex]\(3x = 2x + 18\)[/tex]
Step 1: Move all [tex]\(x\)[/tex] terms to one side.
[tex]\[ 3x - 2x = 18 \][/tex]
Step 2: Simplify.
[tex]\[ x = 18 \][/tex]
So, the solution is [tex]\( x = 18 \)[/tex].
2. [tex]\(5t - 3 = 3t - 5\)[/tex]
Step 1: Move all [tex]\(t\)[/tex] terms to one side and constants to the other side.
[tex]\[ 5t - 3t = -5 + 3 \][/tex]
Step 2: Simplify.
[tex]\[ 2t = -2 \][/tex]
[tex]\[ t = -1 \][/tex]
So, the solution is [tex]\( t = -1 \)[/tex].
3. [tex]\(5x + 9 = 5 + 3x\)[/tex]
Step 1: Move all [tex]\(x\)[/tex] terms to one side and constants to the other side.
[tex]\[ 5x - 3x = 5 - 9 \][/tex]
Step 2: Simplify.
[tex]\[ 2x = -4 \][/tex]
[tex]\[ x = -2 \][/tex]
So, the solution is [tex]\( x = -2 \)[/tex].
4. [tex]\(4z + 3 = 6 + 2z\)[/tex]
Step 1: Move all [tex]\(z\)[/tex] terms to one side and constants to the other side.
[tex]\[ 4z - 2z = 6 - 3 \][/tex]
Step 2: Simplify.
[tex]\[ 2z = 3 \][/tex]
[tex]\[ z = \frac{3}{2} \][/tex]
So, the solution is [tex]\( z = \frac{3}{2} \)[/tex].
5. [tex]\(2x - 1 = 14 - x\)[/tex]
Step 1: Move all [tex]\(x\)[/tex] terms to one side and constants to the other side.
[tex]\[ 2x + x = 14 + 1 \][/tex]
Step 2: Simplify.
[tex]\[ 3x = 15 \][/tex]
[tex]\[ x = 5 \][/tex]
So, the solution is [tex]\( x = 5 \)[/tex].
6. [tex]\(8x + 4 = 3(x - 1) + 7\)[/tex]
Step 1: Expand and simplify.
[tex]\[ 8x + 4 = 3x - 3 + 7 \][/tex]
[tex]\[ 8x + 4 = 3x + 4 \][/tex]
Step 2: Move all [tex]\(x\)[/tex] terms to one side.
[tex]\[ 8x - 3x = 4 - 4 \][/tex]
[tex]\[ 5x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
So, the solution is [tex]\( x = 0 \)[/tex].
7. [tex]\(x = \frac{4}{5}(x + 10)\)[/tex]
Step 1: Distribute the factor on the right-hand side.
[tex]\[ x = \frac{4}{5}x + 8 \][/tex]
Step 2: Move all [tex]\(x\)[/tex] terms to one side.
[tex]\[ x - \frac{4}{5}x = 8 \][/tex]
[tex]\[ \frac{1}{5}x = 8 \][/tex]
Step 3: Simplify.
[tex]\[ x = 8 \cdot 5 \][/tex]
[tex]\[ x = 40 \][/tex]
So, the solution is [tex]\( x = 40 \)[/tex].
8. [tex]\(\frac{2x}{3} + 1 = \frac{7x}{15} + 3\)[/tex]
Step 1: Clear the fractions by finding a common denominator (15).
[tex]\[ 10x + 15 = 7x + 45 \][/tex]
Step 2: Move all [tex]\(x\)[/tex] terms to one side and constants to the other side.
[tex]\[ 10x - 7x = 45 - 15 \][/tex]
[tex]\[ 3x = 30 \][/tex]
[tex]\[ x = 10 \][/tex]
So, the solution is [tex]\( x = 10 \)[/tex].
9. [tex]\(2y + \frac{5}{3} = \frac{26}{3} - y\)[/tex]
Step 1: Multiply through by 3 to clear the fractions.
[tex]\[ 6y + 5 = 26 - 3y \][/tex]
Step 2: Move all [tex]\(y\)[/tex] terms to one side and constants to the other side.
[tex]\[ 6y + 3y = 26 - 5 \][/tex]
[tex]\[ 9y = 21 \][/tex]
[tex]\[ y = \frac{21}{9} \][/tex]
[tex]\[ y = \frac{7}{3} \][/tex]
[tex]\[ y = 2.333 \][/tex]
So, the solution is [tex]\( y = 2.333 \)[/tex].
10. [tex]\(3m = 5m - \frac{8}{5}\)[/tex]
Step 1: Move all [tex]\(m\)[/tex] terms to one side.
[tex]\[ 3m - 5m = - \frac{8}{5} \][/tex]
[tex]\[ -2m = - \frac{8}{5} \][/tex]
Step 2: Simplify and solve for [tex]\(m\)[/tex].
[tex]\[ m = \frac{8}{10} \][/tex]
[tex]\[ m = 0.8 \][/tex]
So, the solution is [tex]\( m = 0.8 \)[/tex].
Thus, the solutions are:
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]\( y = 2.333 \)[/tex]
10. [tex]\( m = 0.8 \)[/tex]
1. [tex]\(3x = 2x + 18\)[/tex]
Step 1: Move all [tex]\(x\)[/tex] terms to one side.
[tex]\[ 3x - 2x = 18 \][/tex]
Step 2: Simplify.
[tex]\[ x = 18 \][/tex]
So, the solution is [tex]\( x = 18 \)[/tex].
2. [tex]\(5t - 3 = 3t - 5\)[/tex]
Step 1: Move all [tex]\(t\)[/tex] terms to one side and constants to the other side.
[tex]\[ 5t - 3t = -5 + 3 \][/tex]
Step 2: Simplify.
[tex]\[ 2t = -2 \][/tex]
[tex]\[ t = -1 \][/tex]
So, the solution is [tex]\( t = -1 \)[/tex].
3. [tex]\(5x + 9 = 5 + 3x\)[/tex]
Step 1: Move all [tex]\(x\)[/tex] terms to one side and constants to the other side.
[tex]\[ 5x - 3x = 5 - 9 \][/tex]
Step 2: Simplify.
[tex]\[ 2x = -4 \][/tex]
[tex]\[ x = -2 \][/tex]
So, the solution is [tex]\( x = -2 \)[/tex].
4. [tex]\(4z + 3 = 6 + 2z\)[/tex]
Step 1: Move all [tex]\(z\)[/tex] terms to one side and constants to the other side.
[tex]\[ 4z - 2z = 6 - 3 \][/tex]
Step 2: Simplify.
[tex]\[ 2z = 3 \][/tex]
[tex]\[ z = \frac{3}{2} \][/tex]
So, the solution is [tex]\( z = \frac{3}{2} \)[/tex].
5. [tex]\(2x - 1 = 14 - x\)[/tex]
Step 1: Move all [tex]\(x\)[/tex] terms to one side and constants to the other side.
[tex]\[ 2x + x = 14 + 1 \][/tex]
Step 2: Simplify.
[tex]\[ 3x = 15 \][/tex]
[tex]\[ x = 5 \][/tex]
So, the solution is [tex]\( x = 5 \)[/tex].
6. [tex]\(8x + 4 = 3(x - 1) + 7\)[/tex]
Step 1: Expand and simplify.
[tex]\[ 8x + 4 = 3x - 3 + 7 \][/tex]
[tex]\[ 8x + 4 = 3x + 4 \][/tex]
Step 2: Move all [tex]\(x\)[/tex] terms to one side.
[tex]\[ 8x - 3x = 4 - 4 \][/tex]
[tex]\[ 5x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
So, the solution is [tex]\( x = 0 \)[/tex].
7. [tex]\(x = \frac{4}{5}(x + 10)\)[/tex]
Step 1: Distribute the factor on the right-hand side.
[tex]\[ x = \frac{4}{5}x + 8 \][/tex]
Step 2: Move all [tex]\(x\)[/tex] terms to one side.
[tex]\[ x - \frac{4}{5}x = 8 \][/tex]
[tex]\[ \frac{1}{5}x = 8 \][/tex]
Step 3: Simplify.
[tex]\[ x = 8 \cdot 5 \][/tex]
[tex]\[ x = 40 \][/tex]
So, the solution is [tex]\( x = 40 \)[/tex].
8. [tex]\(\frac{2x}{3} + 1 = \frac{7x}{15} + 3\)[/tex]
Step 1: Clear the fractions by finding a common denominator (15).
[tex]\[ 10x + 15 = 7x + 45 \][/tex]
Step 2: Move all [tex]\(x\)[/tex] terms to one side and constants to the other side.
[tex]\[ 10x - 7x = 45 - 15 \][/tex]
[tex]\[ 3x = 30 \][/tex]
[tex]\[ x = 10 \][/tex]
So, the solution is [tex]\( x = 10 \)[/tex].
9. [tex]\(2y + \frac{5}{3} = \frac{26}{3} - y\)[/tex]
Step 1: Multiply through by 3 to clear the fractions.
[tex]\[ 6y + 5 = 26 - 3y \][/tex]
Step 2: Move all [tex]\(y\)[/tex] terms to one side and constants to the other side.
[tex]\[ 6y + 3y = 26 - 5 \][/tex]
[tex]\[ 9y = 21 \][/tex]
[tex]\[ y = \frac{21}{9} \][/tex]
[tex]\[ y = \frac{7}{3} \][/tex]
[tex]\[ y = 2.333 \][/tex]
So, the solution is [tex]\( y = 2.333 \)[/tex].
10. [tex]\(3m = 5m - \frac{8}{5}\)[/tex]
Step 1: Move all [tex]\(m\)[/tex] terms to one side.
[tex]\[ 3m - 5m = - \frac{8}{5} \][/tex]
[tex]\[ -2m = - \frac{8}{5} \][/tex]
Step 2: Simplify and solve for [tex]\(m\)[/tex].
[tex]\[ m = \frac{8}{10} \][/tex]
[tex]\[ m = 0.8 \][/tex]
So, the solution is [tex]\( m = 0.8 \)[/tex].
Thus, the solutions are:
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]\( y = 2.333 \)[/tex]
10. [tex]\( m = 0.8 \)[/tex]
