Sure, let's go through the steps needed to solve the equation [tex]\(\frac{2x + 6}{5} = 4x - 3\)[/tex] and prove that [tex]\(x = \frac{7}{6}\)[/tex].
Proof:
1. Equation Given: [tex]\(\frac{2x + 6}{5} = 4x - 3\)[/tex]
- Reason: Given in the problem.
2. Remove the Fraction by Multiplying both Sides by 5:
[tex]\[
5 \cdot \frac{2x + 6}{5} = 5 \cdot (4x - 3)
\][/tex]
- Reason: To eliminate the fraction and simplify the equation.
3. Simplify the left side:
[tex]\[
2x + 6 = 5 \cdot 4x - 5 \cdot 3
\][/tex]
- Reason: Multiplication property of equality.
4. Simplify further:
[tex]\[
2x + 6 = 20x - 15
\][/tex]
- Reason: Multiply the numbers.
5. Move terms involving [tex]\(x\)[/tex] to one side:
[tex]\[
2x - 20x = -15 - 6
\][/tex]
- Reason: Subtract [tex]\(20x\)[/tex] from both sides and subtract [tex]\(6\)[/tex] from both sides.
6. Simplify the equation:
[tex]\[
-18x = -21
\][/tex]
- Reason: Combining like terms.
7. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-21}{-18}
\][/tex]
- Reason: Divide both sides by -18.
8. Simplify the fraction:
[tex]\[
x = \frac{21}{18} = \frac{7}{6}
\][/tex]
- Reason: Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (3).
Therefore, the solution to the equation [tex]\(\frac{2x + 6}{5} = 4x - 3\)[/tex] is indeed [tex]\(x = \frac{7}{6}\)[/tex].
