Absolutely! Let's go through the solution step-by-step to solve the equation:
[tex]\[ \frac{3x}{5} - \frac{4x}{15} = 7 - \frac{2x - 2}{15} \][/tex]
### Step 1: Simplify the Equation
First, we want to combine the fractions on both sides of the equation.
Consider the left-hand side:
[tex]\[ \frac{3x}{5} - \frac{4x}{15} \][/tex]
To combine these, we need a common denominator. The least common multiple of [tex]\( 5 \)[/tex] and [tex]\( 15 \)[/tex] is [tex]\( 15 \)[/tex]. So, convert the fractions to have the same denominator:
[tex]\[ \frac{3x}{5} = \frac{3x \cdot 3}{5 \cdot 3} = \frac{9x}{15} \][/tex]
Now, combine the fractions:
[tex]\[ \frac{9x}{15} - \frac{4x}{15} = \frac{9x - 4x}{15} = \frac{5x}{15} = \frac{x}{3} \][/tex]
So, the equation becomes:
[tex]\[ \frac{x}{3} = 7 - \frac{2x - 2}{15} \][/tex]
### Step 2: Simplify the Right-Hand Side
Now, let's look at the right-hand side of the equation:
[tex]\[ 7 - \frac{2x - 2}{15} \][/tex]
We need to combine these terms into a single fraction. Start by expressing [tex]\( 7 \)[/tex] with the same denominator:
[tex]\[ 7 = \frac{7 \cdot 15}{15} = \frac{105}{15} \][/tex]
So, the equation now is:
[tex]\[ \frac{x}{3} = \frac{105}{15} - \frac{2x - 2}{15} \][/tex]
Combine the fractions on the right-hand side:
[tex]\[ \frac{105 - (2x - 2)}{15} = \frac{105 - 2x + 2}{15} = \frac{107 - 2x}{15} \][/tex]
Thus, the equation becomes:
[tex]\[ \frac{x}{3} = \frac{107 - 2x}{15} \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
To clear the fractions, multiply every term by the common denominator, which is [tex]\( 15 \)[/tex]:
[tex]\[ 15 \cdot \frac{x}{3} = 15 \cdot \frac{107 - 2x}{15} \][/tex]
This simplifies to:
[tex]\[ 5x = 107 - 2x \][/tex]
Combine the [tex]\( x \)[/tex] terms on one side:
[tex]\[ 5x + 2x = 107 \][/tex]
[tex]\[ 7x = 107 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{107}{7} = 15 + \frac{2}{7} = \frac{741 + 2}{49} = \frac{743}{49} \][/tex]
So, the solution simplifies to:
[tex]\[ x = \frac{107}{7} \][/tex]
### Solution to the Equation
Thus, the solution to the given equation [tex]\(\frac{3 x}{5}-\frac{4 x}{15}=7-\frac{2 x-2}{15}\)[/tex] is:
[tex]\[ x = \frac{107}{7} \][/tex]
