Certainly! Let me provide you with a detailed, step-by-step explanation along with justifications for solving the given equation [tex]\( 5(2x+1)=2(4x-7)+3 \)[/tex].
The completed table would be:
\begin{tabular}{|c|c|}
\hline step & Justifications \\
\hline [tex]$5(2 x+1)=2(4 x-7)+3$[/tex] & given \\
\hline [tex]$10 x+5=8 x-14+3$[/tex] & distributive property \\
\hline [tex]$10 x+5=8 x-11$[/tex] & combine like terms \\
\hline [tex]$2 x+5=-11$[/tex] & addition property of equality \\
\hline [tex]$2 x=-16$[/tex] & addition property of equality \\
\hline [tex]$x=-8$[/tex] & multiplication property of equality \\
\hline
\end{tabular}
Here is the step-by-step solution along with justifications:
1. Start with the given equation:
[tex]\[
5(2x+1) = 2(4x-7) + 3
\][/tex]
Justification: This is given.
2. Apply the distributive property:
[tex]\[
5 \cdot 2x + 5 \cdot 1 = 2 \cdot 4x + 2 \cdot (-7) + 3
\][/tex]
After simplifying, we get:
[tex]\[
10x + 5 = 8x - 14 + 3
\][/tex]
Justification: Distributive property.
3. Combine like terms on the right side:
[tex]\[
10x + 5 = 8x - 11
\][/tex]
Justification: Combine like terms.
4. Subtract [tex]\(8x\)[/tex] from both sides of the equation to start isolating [tex]\(x\)[/tex]:
[tex]\[
10x - 8x + 5 = -11
\][/tex]
Simplifying, we get:
[tex]\[
2x + 5 = -11
\][/tex]
Justification: Addition property of equality (subtracting [tex]\(8x\)[/tex] from both sides).
5. Next, subtract [tex]\(5\)[/tex] from both sides to further isolate [tex]\(x\)[/tex]:
[tex]\[
2x + 5 - 5 = -11 - 5
\][/tex]
Simplifying, we get:
[tex]\[
2x = -16
\][/tex]
Justification: Addition property of equality (subtracting [tex]\(5\)[/tex] from both sides).
6. Finally, divide both sides by [tex]\(2\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-16}{2}
\][/tex]
Simplifying, we get:
[tex]\[
x = -8
\][/tex]
Justification: Multiplication property of equality (dividing both sides by [tex]\(2\)[/tex]).
Thus, the solution to the equation [tex]\(5(2x+1) = 2(4x-7) + 3\)[/tex] is [tex]\(x = -8\)[/tex].
