By Nicole Harms, Updated Aug 30, 2022

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Division in algebraic equations often feels intimidating, especially when variables like n and x appear. By breaking a problem down into manageable steps, you can tackle even the most complex equations with confidence.

Step 1 – Write the Equation Clearly

Copy your equation onto a separate sheet. For our first example, we’ll use:

\( \frac{3n}{5}=12 \)

Step 2 – Eliminate the Denominator

To isolate the variable, first remove the division by the constant. Multiply both sides by the denominator (5 in this case):

\( \frac{3n}{5}\times5 = 12\times5 \)

This simplifies to:

\( 3n = 60 \)

Step 3 – Isolate the Variable

Next, divide both sides by the coefficient of the variable (3):

\( \frac{3n}{3} = \frac{60}{3} \)

Yielding:

\( n = 20 \)

Step 4 – Verify Your Result

Check by substituting back into the original equation:

\( \frac{3\times20}{5} = 12 \)

Since the equality holds, the solution is correct.

Step 5 – Tackle More Complex Equations

Apply the same strategy to a more involved example:

\( \frac{48x^2+4x-70}{6x-7}=90 \)

Step 6 – Factor Both Numerator and Denominator

Factor the numerator fully. Here it becomes:

\( (8x+10)(6x-7) \)

The denominator is already simplified.

Step 7 – Cancel Common Factors

Since \(6x-7\) appears in both the numerator and denominator, it cancels out, leaving:

\( 8x+10 = 90 \)

Now solve for x:

\( 8x = 80 \)

\( x = 10 \)

Step 8 – Confirm the Solution

Substitute back to verify:

\( \frac{48\times10^2+4\times10-70}{6\times10-7}= \frac{4770}{53}=90 \)

Things Needed

• Paper
• Pencil

TL;DR (Too Long; Didn’t Read)

Always factor an equation completely before isolating the variable. If a common factor exists—like the 6 in 6x+12—factor it out first, e.g., 6(x+2). This simplifies subsequent steps.

Warning

When manipulating an equation, perform the same operation on both sides. If you divide one side by 2, you must divide the other side by 2 as well.