Linear Equations
Linear equations form the foundation of algebra, yet 8th graders consistently struggle with the systematic approach required to isolate variables. When students face 3x + 7 = 22, they often jump to mental math instead of following the step-by-step process that builds algebraic thinking.
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Why it matters
Linear equations appear everywhere from calculating phone bills to determining break-even points in business. A contractor estimating materials uses equations like 15x + 200 = 800 to find how many hours of labor they can afford within budget. Engineers design bridges using linear relationships between load and stress. Financial planners use linear equations to project savings growth over time. CCSS.8.EE and CCSS.HSA.REI standards emphasize these real-world connections because students who master linear equations gain problem-solving tools for physics, chemistry, economics, and countless career paths. The systematic thinking required to solve 6x + 28 = 2x + 4 translates directly to breaking down complex problems into manageable steps across all STEM fields.
How to solve linear equations
Linear equations — how to
- Collect x-terms on one side, constants on the other.
- Do the same operation to both sides (add, subtract, multiply, divide).
- Divide by the coefficient of x to isolate x.
Example: 3x + 7 = 22 → 3x = 15 → x = 5.
Worked examples
x + 3 = 10
Answer: x = 7
- Subtract 3 from both sides → x = 10 − 3 — To isolate x, undo the addition.
- Calculate → x = 7 — 10 − 3 = 7.
- Verify → 7 + 3 = 10 ✓ — Substitution confirms the solution.
7x + 2 = -54
Answer: x = -8
- Subtract 2 from both sides → 7x = -56 — Isolate the x term by removing the constant.
- Divide both sides by 7 → x = -8 — -56 ÷ 7 = -8.
- Verify → 7(-8) + 2 = -54 ✓ — Substitution confirms the solution.
6x + 28 = 2x + 4
Answer: x = -6
- Subtract 2x from both sides → 4x + 28 = 4 — Collect all x terms on one side.
- Subtract 28 from both sides → 4x = -24 — Move constants to the other side.
- Divide both sides by 4 → x = -6 — -24 ÷ 4 = -6.
- Verify → LHS = RHS = -8 ✓ — Both sides equal the same value.
Common mistakes
- ✗Students often subtract only from one side, writing 3x + 7 = 22 as 3x = 22 instead of 3x = 15, forgetting that operations must be performed on both sides of the equation.
- ✗When collecting like terms in 6x + 28 = 2x + 4, students incorrectly combine unlike terms, writing 8x + 32 = 0 instead of properly moving variables to one side first.
- ✗Students frequently divide incorrectly when the coefficient is negative, solving -4x = 12 as x = 3 instead of x = -3, missing the sign change.
- ✗With fractional coefficients like (2/3)x = 8, students often multiply by the numerator only, getting x = 16 instead of multiplying by the reciprocal 3/2 to get x = 12.
Practice on your own
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