Problem 27 - Algebra for College Students

Question

Find the solution set to each equation. $$\frac{a}{3}=\frac{-1}{4}$$

Step-by-Step Solution

Answer

The solution set is \(a = -\frac{3}{4}\).

Step 1: Isolate the Variable

Start by isolating the variable on one side of the equation. In this case, multiply both sides by 3 to eliminate the fraction with the variable \(a\). \[ \frac{a}{3} \times 3 = \frac{-1}{4} \times 3 \] This simplifies to: \[ a = -\frac{3}{4} \]

Step 2: Verify the Solution

Substitute \(a = -\frac{3}{4}\) back into the original equation to verify if the solution is correct. \[ \frac{-3/4}{3} = \frac{-1}{4} \] Simplify the left side: \[ -\frac{3}{4} \times \frac{1}{3} = \frac{-1}{4} \] Both sides are equal, confirming that the solution is correct.

Key Concepts

isolate the variable

Isolating the variable is a method used in algebra to solve equations for an unknown. In our example, we started with the equation \( \frac{a}{3} = \frac{-1}{4} \). To isolate the variable, we need to get rid of the fraction containing \( a \). This means we want \( a \) on one side of the equation by itself.
We can do this by multiplying both sides of the equation by 3, the denominator of the fraction with the variable. This action eliminates the fraction because the 3s will cancel out each other.
Here's the math:
\ \frac{a}{3} \times 3 = \frac{-1}{4} \times 3 \
Simplifying, we get:
\ a = -\frac{3}{4} \
Now, the variable \( a \) is isolated on the left side of the equation. This is a crucial step in solving any algebraic equation because it clearly shows the value of the unknown.

verify the solution

Once you have isolated the variable, it's important to verify that your solution is correct. Verification involves substituting your solution back into the original equation to see if both sides are equal.
In our case, we found that \( a = -\frac{3}{4} \). Let's substitute it back into the original equation:
\ \frac{-3}{4} \div 3 = \frac{-1}{4} \
This simplifies to:
\ -\frac{3}{4} \times \frac{1}{3} = \frac{-1}{4} \
When simplified further:
The left side becomes:
\ \frac{- 3 \times 1}{4 \times 3} = \frac{-1}{4} \
Both sides of the equation are equal, which confirms that the solution, \( a = -\frac{3}{4} \), is correct. Verification is a vital step to ensure your solution is accurate and satisfies the original equation.

eliminate fractions

Fractions can make solving equations more complex, but eliminating them often simplifies the process. The key is to find a common denominator or multiply through by the denominators to get rid of the fractions.
In our example, the equation was:
\ \frac{a}{3} = \frac{-1}{4} \
The fraction involving \( a \) has a denominator of 3. To eliminate this fraction, we multiplied both sides by 3:
\ \frac{a}{3} \times 3 = \frac{-1}{4} \times 3 \
This step removed the fraction from the variable side, isolating \( a \) and simplifying the right side:
\ a = -\frac{3}{4} \
By eliminating fractions, the equation becomes much easier to handle and solve. It is often one of the first steps in simplifying an algebraic equation.