Problem 52 - Algebra for College Students

Question

Suppose that in solving the following equation, we multiply each side by \(12,\) rather than the least common denominator, \(6 .\) Would we obtain the correct solution? Explain. $$ \frac{1}{3} x+\frac{1}{2} x=\frac{1}{6} x $$

Step-by-Step Solution

Answer

Yes, multiplying by 12 gives the correct solution, which is \( x = 0 \).

Step 1: Original Equation

Consider the given equation: \[ \frac{1}{3} x + \frac{1}{2} x = \frac{1}{6} x \]

Step 2: Multiply Both Sides by 12

To eliminate the denominators, multiply every term in the equation by 12: \[ 12 \left( \frac{1}{3} x \right) + 12 \left( \frac{1}{2} x \right) = 12 \left( \frac{1}{6} x \right) \]Simplify each term: \[ 4x + 6x = 2x \]

Step 3: Simplify the Equation

Combine like terms on the left side: \[ 4x + 6x = 10x \]Now the equation is: \[ 10x = 2x \]

Step 4: Isolate the Variable

Subtract 2x from both sides to isolate x: \[ 10x - 2x = 0 \]which simplifies to: \[ 8x = 0 \]

Step 5: Solve for x

Divide both sides by 8: \[ x = \frac{0}{8} \]Therefore, \[ x = 0 \]

Step 6: Verify the Solution

Substituting \( x = 0 \) back into the original equation: \[ \frac{1}{3} (0) + \frac{1}{2} (0) = \frac{1}{6} (0) \]All terms on both sides equal 0, so the original equation holds true.

Key Concepts

Multiplying Both Sides of an Equation

In algebra, one of the golden rules is that you can perform the same operation on both sides of an equation without changing its equality. This is especially helpful when dealing with fractions in equations. For instance, in the equation \(\frac{1}{3}x + \frac{1}{2}x = \frac{1}{6}x\), if we multiply every term by 12, we effectively eliminate the denominators. This makes the equation easier to solve. Here’s what happens:
First, distribute the 12 to each term:
\[ 12 \left( \frac{1}{3}x \right) + 12 \left( \frac{1}{2}x \right) = 12 \left( \frac{1}{6}x \right) \]
Simplify each term by performing the multiplication:
\[ 4x + 6x = 2x \]
By multiplying both sides by the same number, we ensure that the equation remains balanced, allowing us to focus on isolating and solving for the variable.

Isolating the Variable

After simplifying the equation \[ 4x + 6x = 2x \], the next step is to combine like terms to further simplify the equation. This gives us:
\[ 10x = 2x \]
The goal here is to isolate the variable, which in this case is x. We do this by getting all the x terms on one side of the equation. Subtract \(2x\) from both sides:
\[ 10x - 2x = 2x - 2x \]
This simplifies to:
\[ 8x = 0 \]
Now, we divide both sides by 8 to solve for x:
\[ x = \frac{0}{8} = 0 \]
By isolating the variable, we find that \(x = 0\). Remember, isolating the variable is crucial because it allows us to determine the value of the unknown in the equation.

Verifying Solutions in Algebra

Once you solve for the variable, it's always a good practice to verify the solution. To do this, substitute the solution back into the original equation and check if both sides are equal. For our example, substituting \(x = 0\):
\[ \frac{1}{3}(0) + \frac{1}{2}(0) = \frac{1}{6}(0) \]
Evaluating both sides, we get:
\[ 0 + 0 = 0 \]
This confirms that the solution \(x = 0\) satisfies the original equation. Verification helps ensure that no mistakes were made during solving. It’s an essential final step in solving linear equations as it validates the correctness of your solution.