Problem 8 - Elementary Algebra

Question

For exercises 1-10, (a) solve. (b) check. $$ \frac{2}{9} x+\frac{5}{3}=\frac{5}{9} x+\frac{7}{3} $$

Step-by-Step Solution

Answer

x = -2

Step 1 - Isolate the variable term

Subtract \( \frac{5}{9} x \) from both sides of the equation to get all the x terms on one side. \[ \frac{2}{9} x - \frac{5}{9} x + \frac{5}{3} = \frac{7}{3} \] Simplify the equation: \[ -\frac{3}{9} x + \frac{5}{3} = \frac{7}{3} \] which simplifies to \[ -\frac{1}{3} x + \frac{5}{3} = \frac{7}{3} \]

Step 2 - Remove the constant term

Subtract \( \frac{5}{3} \) from both sides to isolate the term involving x: \[ -\frac{1}{3} x + \frac{5}{3} - \frac{5}{3} = \frac{7}{3} - \frac{5}{3} \] Simplify the equation: \[ -\frac{1}{3} x = \frac{2}{3} \]

Step 3 - Solve for x

Multiply both sides by \( -3 \) to solve for x: \[ x = -2 \]

Step 4 - Check the solution

Substitute \( x = -2 \) back into the original equation: \[ \frac{2}{9}(-2) + \frac{5}{3} = \frac{5}{9}(-2) + \frac{7}{3} \] Simplify both sides: \[ -\frac{4}{9} + \frac{15}{9} = -\frac{10}{9} + \frac{21}{9} \] which simplifies to \[ \frac{11}{9} = \frac{11}{9} \] The left-hand side equals the right-hand side, confirming the solution as correct.

Key Concepts

Isolating Variable Terms

Isolating the variable term is a crucial first step in solving linear equations. The goal is to get all the terms containing the variable on one side of the equation. In our example, we start with:
$$ \frac{2}{9} x + \frac{5}{3} = \frac{5}{9} x + \frac{7}{3} $$
To isolate the variable term, we subtract \(\frac{5}{9} x\) from both sides. This leaves all the x terms on one side and the constants on the other side:
\[ \frac{2}{9} x - \frac{5}{9} x + \frac{5}{3} = \frac{7}{3} \]
Perform the subtraction:
\[ -\frac{3}{9} x + \frac{5}{3} = \frac{7}{3} \]
Which simplifies further to:
\[ -\frac{1}{3} x + \frac{5}{3} = \frac{7}{3} \]
With all x terms on one side, we're ready for the next step.

Simplifying Equations

Simplifying equations helps make them more manageable to solve. Once the variable terms are isolated, we need to simplify by removing constants from the variable side. For our equation, we have:
\[ -\frac{1}{3} x + \frac{5}{3} = \frac{7}{3} \]
To eliminate the \(\frac{5}{3}\) term, subtract \(\frac{5}{3}\) from both sides:
\