Sure, let's clearly write down each step of the algebraic proof for the equation [tex]\(\frac{2x + 6}{5} = 4x - 3\)[/tex]:
### Proof:
#### Statements:
1. [tex]\(\frac{2x + 6}{5} = 4x - 3\)[/tex]
2. [tex]\(2x + 6 = 20x - 15\)[/tex]
3. [tex]\(6 + 15 = 20x - 2x\)[/tex]
4. [tex]\(21 = 18x\)[/tex]
5. [tex]\(x = \frac{21}{18}\)[/tex]
6. [tex]\(x = \frac{7}{6}\)[/tex]
#### Reasons:
1. Given equation.
2. Multiply both sides of the equation by 5 to eliminate the denominator.
3. Add 15 to both sides to eliminate the negative constant on the right.
4. Combine the [tex]\(x\)[/tex] terms on the right side by subtracting [tex]\(2x\)[/tex] from both sides.
5. Divide both sides by 18 to solve for [tex]\(x\)[/tex].
6. Simplify the fraction [tex]\(\frac{21}{18}\)[/tex] to get [tex]\(\frac{7}{6}\)[/tex].
### Detailed Steps:
1. Start with the given equation:
[tex]\[
\frac{2x + 6}{5} = 4x - 3
\][/tex]
2. Multiply both sides by 5 to clear the fraction:
[tex]\[
5 \cdot \frac{2x + 6}{5} = 5 \cdot (4x - 3)
\][/tex]
Simplifying both sides:
[tex]\[
2x + 6 = 20x - 15
\][/tex]
3. Isolate the variable by moving all [tex]\(x\)[/tex]-terms to one side:
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[
2x + 6 - 2x = 20x - 2x - 15
\][/tex]
Simplifying both sides:
[tex]\[
6 = 18x - 15
\][/tex]
4. Move the constant term to the other side:
Add 15 to both sides of the equation:
[tex]\[
6 + 15 = 18x - 15 + 15
\][/tex]
Simplifying both sides:
[tex]\[
21 = 18x
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Divide both sides by 18:
[tex]\[
\frac{21}{18} = x
\][/tex]
Simplifying the fraction:
[tex]\[
x = \frac{7}{6}
\][/tex]
Thus, we have shown through algebraic manipulation that the solution to the equation [tex]\(\frac{2x + 6}{5} = 4x - 3\)[/tex] is indeed [tex]\(x = \frac{7}{6}\)[/tex].
