Solving a two-step linear equation and checking the solution

Key takeaway: This equation is solved by distributing both sides, collecting variables on one side, and isolating x with inverse operations.

Set up the equation

We need to solve

$5(2x+3)=4(x+1).$

This is a linear equation, so the goal is to remove parentheses first, then gather the $x$-terms on one side and the constants on the other.

Distribute and simplify

Expand both sides:

$10x+15=4x+4.$

Now move the variable terms to one side by subtracting $4x$ from both sides:

$6x+15=4.$

Next subtract 15 from both sides:

$6x=-11.$

Finally divide by 6:

$x=-\frac{11}{6}.$

Check the answer

Substitute back into the original equation:

Left side:

=5\left(-\frac{11}{3}+3\right) =5\left(-\frac{2}{3}\right) =-\frac{10}{3}.$$ Right side: $$4\left(-\frac{11}{6}+1\right) =4\left(-\frac{5}{6}\right) =-\frac{10}{3}.$$ Both sides match, so the solution is correct. ## Match the choice The solution is $$\boxed{x=-\frac{11}{6}}$$ which corresponds to option **(b)**. ## Common equation-solving habit A reliable habit is to keep the variable on one side and the constants on the other. That makes the algebra organized and helps avoid sign mistakes. Checking the solution at the end is also important, especially on multiple-choice questions where nearby fractions can be tempting but incorrect. --- **Pitfalls the pros know** 👇 A common error is to distribute only the first term on each side and forget the constant term. Another is to move $4x$ across the equals sign but change the sign incorrectly, which can produce a wrong coefficient such as $14x$ or $2x$ instead of $6x$. Some students also stop after finding a fraction and never substitute it back into the original equation. For linear equations, a quick check is worthwhile because it confirms both the algebra and the choice selection. **What if the problem changes?** If the equation were changed to $5(2x+3)=4(x-1)$, the distribution step would become $10x+15=4x-4$, leading to $6x=-19$ and $x=-19/6$. If it were written as $5(2x-3)=4(x+1)$, the answer would change again because the constant term on the left becomes negative. These small sign changes alter the final fraction, so each parenthesis must be expanded exactly as written. `Tags`: distributive property, inverse operations, linear equation