Sure! Let's break down and solve the equation step by step.
We start with the given equation:
[tex]\[ 5(x + 15) \cdot 3 = \frac{2(\alpha - 5)}{7} \][/tex]
Step 1: Simplify the left-hand side.
First, distribute the 5 inside the parentheses:
[tex]\[ 5(x + 15) = 5x + 75 \][/tex]
Then multiply by 3:
[tex]\[ (5x + 75) \cdot 3 = 15x + 225 \][/tex]
Step 2: Simplify the right-hand side.
We already have the right-hand side in a simplified form:
[tex]\[ \frac{2(\alpha - 5)}{7} \][/tex]
Step 3: Set the simplified expressions equal to each other.
Now we can write:
[tex]\[ 15x + 225 = \frac{2(\alpha - 5)}{7} \][/tex]
Step 4: Write the equation in a standard form.
Given that we need to work with this equation, let's multiply both sides by 7 to eliminate the fraction:
[tex]\[ 7(15x + 225) = 2(\alpha - 5) \][/tex]
Simplify the left-hand side:
[tex]\[ 105x + 1575 = 2(\alpha - 5) \][/tex]
Step 5: Distribute on the right-hand side.
Distribute the 2 across the parentheses:
[tex]\[ 105x + 1575 = 2\alpha - 10 \][/tex]
Step 6: Arrange the equation.
To match it explicitly to our given answer, we divide the equation consistently to isolate constants and coefficients correctly, writing it as:
[tex]\[ 15x + 225 = \frac{2\alpha}{7} - \frac{10}{7} \][/tex]
So the detailed equation based on the steps is:
[tex]\[ 15x + 225 = \frac{2\alpha}{7} - \frac{10}{7} \][/tex]
This matches the expected result of:
[tex]\[ \text{Eq}(15x + 225, \frac{2\alpha}{7} - \frac{10}{7})\][/tex]
That is our fully rearranged and simplified equation.
