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MathChapter 2: Solving Linear Equations

Notes

Solving Equations with Variables on Both Sides Notes

Collect variable terms strategically, preserve equality through legal operations, and solve equations with grouping, fractions, decimals, and negative coefficients.

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When a variable appears on both sides, the goal is to create an equivalent equation with variable terms on only one side. Nothing actually jumps across the equal sign: you add or subtract the same variable term from both sides, then continue isolating the variable as usual.

Learning objectives

  • Simplify each side independently using distribution and like terms.
  • Use equality properties to collect variable terms on one side and constants on the other.
  • Choose a side strategically to reduce negative arithmetic when possible.
  • Solve equations containing fractions, decimals, parentheses, and negative coefficients.
  • Verify the result in the original equation and diagnose illegal term movement.

The consistent six-step procedure

Distribute → combine → collect variables → collect constants → solve → verify

  1. Distribute

    Remove grouping on either side, carrying the sign of each outside factor.

  2. Combine like terms

    Simplify the left and right sides separately. Never combine across \(=\).

  3. Collect variable terms

    Add or subtract the same variable term on both sides so that one side has no variable term.

  4. Collect constants

    Use addition or subtraction on both sides to place constants opposite the variable term.

  5. Isolate

    Divide by the remaining nonzero coefficient or multiply by its reciprocal.

  6. Verify

    Substitute into the original equation and confirm that both sides match.

Two legal paths, one solution

Collect on the left

From \(7x+4=3x+24\), subtract \(3x\): \(4x+4=24\), leading to \(x=5\).

Collect on the right

Subtract \(7x\): \(4=-4x+24\), which also leads to \(-20=-4x\) and \(x=5\).

Integer coefficients and checking

Worked example

Collect variable terms deliberately

Solve \(8x-13=3x+22\).

  1. Collect variables

    Subtract \(3x\) from both sides: \(5x-13=22\).

  2. Collect constants

    Add \(13\) to both sides: \(5x=35\).

  3. Isolate

    Divide by \(5\): \(x=7\).

  4. Verify

    Left: \(8(7)-13=43\). Right: \(3(7)+22=43\).

The solution is \(x=7\).

Distribution and like terms on both sides

Worked example

Simplify before collecting

Solve \(3(2x-5)+4=2(x+7)+9\).

  1. Distribute

    Write \(6x-15+4=2x+14+9\).

  2. Combine

    Simplify each side: \(6x-11=2x+23\).

  3. Collect variables

    Subtract \(2x\): \(4x-11=23\).

  4. Collect constants

    Add \(11\): \(4x=34\).

  5. Isolate

    Divide by \(4\): \(x=\frac{17}{2}\).

  6. Verify

    Substitution makes both original sides equal \(40\).

The solution is \(x=\frac{17}{2}\).

Fractions and decimals

Clear denominators only when it simplifies the work, and multiply every term on both sides. For decimals, multiplying the whole equation by \(10\), \(100\), or another power of ten creates an equivalent equation with integer coefficients.

Worked example

Variables on both sides with fractions

Solve \(\frac{x}{3}+\frac12=\frac{x}{6}+\frac52\).

  1. Clear denominators

    Multiply every term by \(6\): \(2x+3=x+15\).

  2. Collect variables

    Subtract \(x\) from both sides: \(x+3=15\).

  3. Collect constants

    Subtract \(3\): \(x=12\).

  4. Verify

    Both original sides equal \(\frac92\).

The solution is \(x=12\).
Worked example

Variables on both sides with decimals

Solve \(0.7x-1.8=0.2x+4.7\).

  1. Clear decimals

    Multiply by \(10\): \(7x-18=2x+47\).

  2. Collect variables

    Subtract \(2x\): \(5x-18=47\).

  3. Collect constants

    Add \(18\): \(5x=65\).

  4. Isolate

    Divide by \(5\): \(x=13\).

  5. Verify

    Both original sides equal \(7.3\).

The solution is \(x=13\).

From verbal statement to solution

Worked example

Translate, then solve

Five more than twice a number is \(9\) less than four times the number. Find the number.

  1. Define

    Let \(n\) be the number.

  2. Translate

    Five more than twice the number is \(2n+5\); nine less than four times it is \(4n-9\).

  3. Create equality

    Write \(2n+5=4n-9\).

  4. Collect

    Subtract \(2n\) and add \(9\): \(14=2n\).

  5. Solve and check

    \(n=7\); both verbal quantities equal \(19\).

The number is \(7\).

Common mistakes

Mini check

Check your understanding

Solve \(5(2x-1)=3(x+4)+11\).

  1. \(x=2\)
  2. \(x=3\)
  3. \(x=4\)
  4. \(x=7\)
Show answer and explanation

Answer: \(x=4\)

Distribute: \(10x-5=3x+12+11=3x+23\). Subtract \(3x\), add \(5\), and divide by \(7\): \(7x=28\Rightarrow x=4\).

Key takeaways

Key takeaways

What to remember

  • Distribute and combine like terms on each side before collecting across the equation.
  • Collect variables by applying the same addition or subtraction to both sides.
  • Either side may hold the variable; choose the cleaner path when possible.
  • Clear fractions or decimals only by multiplying every term on both sides.
  • Never combine terms across the equal sign without first using an equality property.
  • Verify the final value in the original equation.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.