Let's go through a detailed, step-by-step algebraic solution to solve the equation [tex]\(\frac{2x + 6}{5} = 4x - 3\)[/tex].
### Statements and Reasons
1. Given Equation:
[tex]\[
\frac{2x + 6}{5} = 4x - 3
\][/tex]
Reason: The problem statement.
2. Clear the fraction by multiplying every term by 5:
[tex]\[
5 \left(\frac{2x + 6}{5}\right) = 5 (4x - 3)
\][/tex]
Simplifying this, we get:
[tex]\[
2x + 6 = 20x - 15
\][/tex]
Reason: Multiplying both sides by 5 to eliminate the fraction.
3. Distribute and combine like terms:
Since the left-hand side remains [tex]\(2x + 6\)[/tex], we rewrite the right-hand side to distribute the 5:
[tex]\[
2x + 6 = 20x - 15
\][/tex]
Reason: Distributing the multiplication on the right-hand side.
4. Isolate the variable by moving [tex]\(2x\)[/tex] to the right-hand side:
Subtract [tex]\(2x\)[/tex] from both sides to keep [tex]\(x\)[/tex] on one side:
[tex]\[
6 = 18x - 15
\][/tex]
Reason: Subtract [tex]\(2x\)[/tex] from both sides to combine like terms.
5. Isolate the constant by moving [tex]\(-15\)[/tex] to the left-hand side:
Add 15 to both sides to move the constant term:
[tex]\[
6 + 15 = 18x
\][/tex]
Simplifying this gives:
[tex]\[
21 = 18x
\][/tex]
Reason: Adding 15 to isolate terms involving [tex]\(x\)[/tex].
6. Solve for [tex]\(x\)[/tex]:
Divide both sides of the equation by 18 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{21}{18}
\][/tex]
Simplify the fraction:
[tex]\[
x = \frac{7}{6}
\][/tex]
Reason: Dividing both sides by 18 to solve for [tex]\(x\)[/tex] and simplifying the fraction.
### Conclusion
The solution to the equation [tex]\(\frac{2x + 6}{5} = 4x - 3\)[/tex] is:
[tex]\[
x = \frac{7}{6}
\][/tex]
This step-by-step process demonstrates that [tex]\(x = \frac{7}{6}\)[/tex] is indeed the correct solution.
