Answer :
Sure, let's solve each equation one by one and check the results.
### 1. Solve [tex]\( 3x = 2x + 18 \)[/tex]
First, let's isolate [tex]\( x \)[/tex]:
[tex]\[ 3x - 2x = 18 \][/tex]
[tex]\[ x = 18 \][/tex]
The solution for this equation is [tex]\( x = 18 \)[/tex].
### 2. Solve [tex]\( 5t - 3 = 3t - 5 \)[/tex]
First, let's get all the [tex]\( t \)[/tex] terms on one side:
[tex]\[ 5t - 3t = -5 + 3 \][/tex]
[tex]\[ 2t = -2 \][/tex]
Now, divide both sides by 2:
[tex]\[ t = -1 \][/tex]
The solution for this equation is [tex]\( t = -1 \)[/tex].
### 3. Solve [tex]\( 4z + 3 = 6 + 2z \)[/tex]
First, let's get all the [tex]\( z \)[/tex] terms on one side:
[tex]\[ 4z - 2z = 6 - 3 \][/tex]
[tex]\[ 2z = 3 \][/tex]
Now, divide both sides by 2:
[tex]\[ z = \frac{3}{2} \][/tex]
The solution for this equation is [tex]\( z = \frac{3}{2} \)[/tex].
### 4. Solve [tex]\( 2x - 1 = 14 - x \)[/tex]
First, let's get all the [tex]\( x \)[/tex] terms on one side:
[tex]\[ 2x + x = 14 + 1 \][/tex]
[tex]\[ 3x = 15 \][/tex]
Now, divide both sides by 3:
[tex]\[ x = 5 \][/tex]
The solution for this equation is [tex]\( x = 5 \)[/tex].
### 5. Solve [tex]\( x = \frac{4}{5}(x + 10) \)[/tex]
First, let's distribute the [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[ x = \frac{4}{5}x + 8 \][/tex]
Now, let's get all the [tex]\( x \)[/tex] terms on one side:
[tex]\[ x - \frac{4}{5}x = 8 \][/tex]
[tex]\[ \frac{1}{5}x = 8 \][/tex]
Finally, multiply both sides by 5:
[tex]\[ x = 40 \][/tex]
The solution for this equation is [tex]\( x = 40 \)[/tex].
### 6. Solve [tex]\( \frac{2x}{3} + 1 = \frac{7x}{15} + 3 \)[/tex]
First, get all the [tex]\( x \)[/tex] terms together:
[tex]\[ \frac{2x}{3} - \frac{7x}{15} = 3 - 1 \][/tex]
To combine the [tex]\( x \)[/tex] terms, we need a common denominator:
[tex]\[ \frac{10x}{15} - \frac{7x}{15} = 2 \][/tex]
[tex]\[ \frac{3x}{15} = 2 \][/tex]
[tex]\[ \frac{x}{5} = 2 \][/tex]
Finally, multiply both sides by 5:
[tex]\[ x = 10 \][/tex]
The solution for this equation is [tex]\( x = 10 \)[/tex].
### 7. Solve [tex]\( 3m = 5m - \frac{8}{5} \)[/tex]
First, get all the [tex]\( m \)[/tex] terms together:
[tex]\[ 3m - 5m = -\frac{8}{5} \][/tex]
[tex]\[ -2m = -\frac{8}{5} \][/tex]
Now, divide both sides by [tex]\(-2\)[/tex]:
[tex]\[ m = \frac{8}{10} \][/tex]
[tex]\[ m = 0.8 \][/tex]
The solution for this equation is [tex]\( m = 0.8 \)[/tex].
So the solutions are:
1. [tex]\( x = 18 \)[/tex]
2. [tex]\( t = -1 \)[/tex]
3. [tex]\( z = \frac{3}{2} \)[/tex]
4. [tex]\( x = 5 \)[/tex]
5. [tex]\( x = 40 \)[/tex]
6. [tex]\( x = 10 \)[/tex]
7. [tex]\( m = 0.8 \)[/tex]
### 1. Solve [tex]\( 3x = 2x + 18 \)[/tex]
First, let's isolate [tex]\( x \)[/tex]:
[tex]\[ 3x - 2x = 18 \][/tex]
[tex]\[ x = 18 \][/tex]
The solution for this equation is [tex]\( x = 18 \)[/tex].
### 2. Solve [tex]\( 5t - 3 = 3t - 5 \)[/tex]
First, let's get all the [tex]\( t \)[/tex] terms on one side:
[tex]\[ 5t - 3t = -5 + 3 \][/tex]
[tex]\[ 2t = -2 \][/tex]
Now, divide both sides by 2:
[tex]\[ t = -1 \][/tex]
The solution for this equation is [tex]\( t = -1 \)[/tex].
### 3. Solve [tex]\( 4z + 3 = 6 + 2z \)[/tex]
First, let's get all the [tex]\( z \)[/tex] terms on one side:
[tex]\[ 4z - 2z = 6 - 3 \][/tex]
[tex]\[ 2z = 3 \][/tex]
Now, divide both sides by 2:
[tex]\[ z = \frac{3}{2} \][/tex]
The solution for this equation is [tex]\( z = \frac{3}{2} \)[/tex].
### 4. Solve [tex]\( 2x - 1 = 14 - x \)[/tex]
First, let's get all the [tex]\( x \)[/tex] terms on one side:
[tex]\[ 2x + x = 14 + 1 \][/tex]
[tex]\[ 3x = 15 \][/tex]
Now, divide both sides by 3:
[tex]\[ x = 5 \][/tex]
The solution for this equation is [tex]\( x = 5 \)[/tex].
### 5. Solve [tex]\( x = \frac{4}{5}(x + 10) \)[/tex]
First, let's distribute the [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[ x = \frac{4}{5}x + 8 \][/tex]
Now, let's get all the [tex]\( x \)[/tex] terms on one side:
[tex]\[ x - \frac{4}{5}x = 8 \][/tex]
[tex]\[ \frac{1}{5}x = 8 \][/tex]
Finally, multiply both sides by 5:
[tex]\[ x = 40 \][/tex]
The solution for this equation is [tex]\( x = 40 \)[/tex].
### 6. Solve [tex]\( \frac{2x}{3} + 1 = \frac{7x}{15} + 3 \)[/tex]
First, get all the [tex]\( x \)[/tex] terms together:
[tex]\[ \frac{2x}{3} - \frac{7x}{15} = 3 - 1 \][/tex]
To combine the [tex]\( x \)[/tex] terms, we need a common denominator:
[tex]\[ \frac{10x}{15} - \frac{7x}{15} = 2 \][/tex]
[tex]\[ \frac{3x}{15} = 2 \][/tex]
[tex]\[ \frac{x}{5} = 2 \][/tex]
Finally, multiply both sides by 5:
[tex]\[ x = 10 \][/tex]
The solution for this equation is [tex]\( x = 10 \)[/tex].
### 7. Solve [tex]\( 3m = 5m - \frac{8}{5} \)[/tex]
First, get all the [tex]\( m \)[/tex] terms together:
[tex]\[ 3m - 5m = -\frac{8}{5} \][/tex]
[tex]\[ -2m = -\frac{8}{5} \][/tex]
Now, divide both sides by [tex]\(-2\)[/tex]:
[tex]\[ m = \frac{8}{10} \][/tex]
[tex]\[ m = 0.8 \][/tex]
The solution for this equation is [tex]\( m = 0.8 \)[/tex].
So the solutions are:
1. [tex]\( x = 18 \)[/tex]
2. [tex]\( t = -1 \)[/tex]
3. [tex]\( z = \frac{3}{2} \)[/tex]
4. [tex]\( x = 5 \)[/tex]
5. [tex]\( x = 40 \)[/tex]
6. [tex]\( x = 10 \)[/tex]
7. [tex]\( m = 0.8 \)[/tex]
