Problem 7 - Elementary Algebra

Question

For exercises 1-10, (a) solve. (b) check. $$ \frac{2}{3} x+\frac{3}{2}=\frac{1}{3} x+\frac{1}{6} $$

Step-by-Step Solution

Answer

-4

Step 1: Combine like terms

First, move all terms involving the variable to one side of the equation. Subtract \(\frac{1}{3}x\) from both sides: \(\frac{2}{3}x - \frac{1}{3}x + \frac{3}{2} = \frac{1}{6}\).

Step 2: Simplify the equation

Simplify the left-hand side: \(\frac{2}{3}x - \frac{1}{3}x = \frac{1}{3}x\), so the equation becomes \(\frac{1}{3}x + \frac{3}{2} = \frac{1}{6}\).

Step 3: Isolate the variable term

Subtract \(\frac{3}{2}\) from both sides to isolate the term with the variable: \(\frac{1}{3}x = \frac{1}{6} - \frac{3}{2}\).

Step 4: Simplify the right-hand side

To subtract the fractions, get a common denominator: \(\frac{1}{6} - \frac{9}{6} = -\frac{8}{6} = -\frac{4}{3}\). So the equation becomes: \(\frac{1}{3}x = -\frac{4}{3}\).

Step 5: Solve for x

Multiply both sides by 3 to solve for \x\: \(x = -4\).

Step 6: Check the solution

Substitute \(x = -4\) back into the original equation to verify the solution: \(\frac{2}{3}(-4) + \frac{3}{2} = \frac{1}{3}(-4) + \frac{1}{6}\). Simplify both sides to confirm they are equal: LHS = -\frac{8}{3} + \frac{9}{6} = -\frac{5}{6}, RHS = -\frac{4}{3} + \frac{1}{6} = -\frac{5}{6}. Since LHS = RHS, \(x = -4\) is correct.

Key Concepts

Combining Like Terms

When solving linear equations, one of the first steps is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our exercise, we need to bring all terms involving the variable to one side of the equation. For example, we subtracted \(\frac{1}{3}x\) from both sides of the equation to combine the x-terms:\( \frac{2}{3}x - \frac{1}{3}x + \frac{3}{2} = \frac{1}{6} \). This is crucial because it simplifies the equation and helps us move towards isolating the variable.

Tips for combining like terms:

• Ensure you move terms with the variable to one side.
• Don't forget to perform the same operation on both sides of the equation.
• Combine constants separately from variable terms.

This step makes the equation more manageable.

Isolating the Variable

After combining like terms, the next step is to isolate the variable. This means getting the variable term alone on one side of the equation. In the exercise, we subtracted \(\frac{3}{2}\) from both sides to isolate \(\frac{1}{3}x\): \(\frac{1}{3}x = \frac{1}{6} - \frac{3}{2} \).

Steps to isolate the variable:

• Use addition or subtraction to move constant terms to the other side of the equation.
• Ensure the variable term remains on one side by itself.
• Continue simplifying until you have the variable alone.
  

After isolating the variable, the equation becomes easier to solve because you're one step away from finding the variable's value.

Fraction Subtraction

Subtracting fractions can be tricky, especially if they have different denominators. In the exercise, we needed to subtract \(\frac{3}{2}\) from \(\frac{1}{6}\). To do this, we first found a common denominator.

\textbf{Steps to subtract fractions:}

• Identify the least common denominator (LCD) of the fractions.
• Convert each fraction to an equivalent fraction using the LCD.
• Subtract the numerators and place the result over the common denominator.

For example, with \(\frac{1}{6} - \frac{3}{2}\), the LCD is 6. Converting, we get \(\frac{1}{6} = \frac{1}{6}\) and \(\frac{3}{2} = \frac{9}{6}\). We can now subtract: \(\frac{1}{6} - \frac{9}{6} = -\frac{8}{6} = -\frac{4}{3}\). This simplification helps solve for the variable more efficiently.

Remember:

• Always find a common denominator first.
• Simplify the fractions if possible after subtraction.
• Double-check your work to avoid mistakes.

Understanding fraction subtraction is crucial for solving equations involving fractions.