How to explain it
At this standard, students solve linear equations with rational number coefficients — fractions and decimals — using LCD-clearing and reciprocal techniques, including equations requiring the distributive property.
The anchor students hold onto: Simplify first. For fractions, multiply every term by the LCD to clear. For decimals, multiply by 10 or 100. Then use inverse operations. Step 1: Define the variable. Step 2: Write the equation. Step 3: Solve. Step 4: Check the answer in the original problem context. Whatever you do to one side, do to the other. Simplify each side FIRST (distribute, then combine), then apply inverse operations in reverse PEMDAS order.
Rational-coefficient fluency feeds directly into 8.F.B.4 (writing and constructing linear functions) and 8.EE.C.8b (solving systems algebraically — next in this bundle).
Worked examples
Example 1
Fraction Coefficient
Solve: (3/4)x + 2 = 5
Step 1Subtract 2 from both sides: (3/4)x = 3.
Step 2Multiply by the reciprocal (4/3): x = 4.
Answerx = 4
Example 2
Decimal Coefficient
Solve: 0.5x + 1.5 = 4
Step 1Subtract 1.5 from both sides: 0.5x = 2.5.
Step 2Divide both sides by 0.5: x = 5.
Answerx = 5
Example 3
Distribute + Fraction
Solve: (1/3)(x + 9) = 5
Step 1Distribute 1/3: (1/3)x + 3 = 5.
Step 2Subtract 3: (1/3)x = 2.
Step 3Multiply by 3: x = 6.
Answerx = 6
Common mistakes
What students write
Multiplying only the variable term by the reciprocal — for example, writing (1/2)x + 4 = 7 → x + 4 = 14 instead of x + 8 = 14.
The fix
Multiply EVERY term on both sides by the LCD or reciprocal.
What students write
Using the original fraction instead of its reciprocal — writing x = 6 × (3/4) instead of x = 6 × (4/3).
The fix
The reciprocal of a/b is b/a — flip numerator and denominator to clear the coefficient.
What students write
Writing an expression for only one quantity instead of the equation for the combined condition — for example, writing "2k + 3 = 39" when "combined they have 39" means both together.
The fix
Read the entire problem: the equation must represent the combined condition, not just one person's count.
What students write
Skipping the variable definition — writing an equation without stating what the variable represents, then solving for the wrong quantity.
The fix
Write "Let ___ = ___" first. The variable must name something specific from the problem context.
Teacher tip
Head off the two predictable errors before they happen. First: Multiply EVERY term on both sides by the LCD or reciprocal. Second: The reciprocal of a/b is b/a — flip numerator and denominator to clear the coefficient.