To solve the equation [tex]\(10x - 16 = \frac{2}{3}(15x + 24)\)[/tex], let's work through it step by step.
1. Expand the right side of the equation:
[tex]\[
10x - 16 = \frac{2}{3} (15x + 24)
\][/tex]
2. Distribute [tex]\(\frac{2}{3}\)[/tex] over [tex]\((15x + 24)\)[/tex]:
[tex]\[
10x - 16 = \frac{2}{3} \cdot 15x + \frac{2}{3} \cdot 24
\][/tex]
[tex]\[
10x - 16 = 10x + 16
\][/tex]
3. Write the equation with fractions evaluated:
[tex]\[
10x - 16 = 10x + 16
\][/tex]
4. Isolate the variable term on one side and constants on the other side:
First, subtract [tex]\(10x\)[/tex] from both sides:
[tex]\[
10x - 10x - 16 = 10x - 10x + 16
\][/tex]
[tex]\[
-16 = 16
\][/tex]
At this point, the left side simplifies to [tex]\(-16\)[/tex], and the right simplifies to [tex]\(16\)[/tex].
5. Analyze the resulting equation:
[tex]\[
10x - 16 = 10x + 16 \Rightarrow -16 ≠ 16
\][/tex]
This resulting equality [tex]\(-16 = 16\)[/tex] is clearly false, meaning there's no value of [tex]\(x\)[/tex] that will make this equation true.
Conclusion: This equation has no solutions. When the variable terms cancel out and leave a false statement like [tex]\(-16 = 16\)[/tex], it indicates that there's no real number [tex]\(x\)[/tex] satisfying the given equation.
