πŸ‡ΊπŸ‡Έ CCSS Math Β· Grade 8

8.EE.C.7: Solving linear equations in one variable

8.EE.C.7 explained: multi-step linear equations with the distributive property, plus one, none or infinitely many solutions. Worked example and practice.

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Common Core standard CCSS.Math.Content.8.EE.C.7

Solve linear equations in one variable.

  • a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
  • b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Grade
Grade 8
Domain
Expressions & Equations (EE)
Cluster
Analyze and solve linear equations and pairs of simultaneous linear equations

Official wording from the Common Core State Standards for Mathematics (Β© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org

What 8.EE.C.7 means

By eighth grade, equations stop being two-step puzzles. Students solve equations such as 3(2x - 5) + 4 = 2x + 9 or 0.5(x + 6) = 1.25x - 3, where the variable appears on both sides and the distributive property and combining like terms must come first. The guiding principle is to keep the equation balanced: whatever is done to one side is done to the other, until x stands alone.

The standard adds a twist that surprises many students. Not every linear equation has exactly one answer. Simplifying 2(x + 3) = 2x + 6 leads to 6 = 6, which is always true, so every value of x works. Simplifying 4x + 1 = 4x - 7 leads to 1 = -7, which is never true, so there is no solution. Students learn to recognize all three outcomes (x = a, a = a, or a = b with a and b different) and to write their own examples of each.

Students should be able to

  • Solve linear equations with variables on both sides, including fraction and decimal coefficients.
  • Use the distributive property and combine like terms before isolating the variable.
  • Recognize when an equation has one solution, no solution or infinitely many solutions.
  • Create an example equation for each of the three solution types.
  • Check a solution by substituting it back into the original equation.

Common misconceptions

Distributing to only the first term

Students rewrite 3(2x - 5) as 6x - 5, multiplying only the first term. The 3 multiplies every term inside the parentheses, giving 6x - 15.

Calling 0 = 0 the answer 'x = 0'

When the variables cancel and a true statement remains, the solution set is every number, not zero. Substituting a few values shows they all work.

Losing the sign when moving terms

Subtracting 4x from both sides of 7 - 4x = 2x + 1 should give 7 = 6x + 1. Students often write 7 = -2x + 1 by treating -4x as +4x.

Stopping at 'no solution' too early

Students sometimes see the same x term on both sides before simplifying and declare no solution. Fully simplify both sides first; the constants decide the outcome.

Worked example: variables on both sides

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

  1. Distribute the 3: 6x - 15 + 4 = 2x + 9.
  2. Combine like terms on the left: 6x - 11 = 2x + 9.
  3. Subtract 2x from both sides: 4x - 11 = 9. Add 11 to both sides: 4x = 20.
  4. Divide by 4: x = 5. Check: 3(10 - 5) + 4 = 19 and 2(5) + 9 = 19.

Answer: x = 5

Teaching 8.EE.C.7

Balance models and algebra tiles help students who still see the equals sign as 'the answer comes next'. For the three solution types, try a card sort where students simplify each equation and then sort by outcome, followed by a challenge to change one number in an equation so it switches from one solution to none.

Assessments commonly include a multiple-choice item asking how many solutions an equation has and multi-step equations with fractions or decimals. Encourage clearing fractions by multiplying both sides by a common denominator, and always finish with a substitution check.

6 practice questions

Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.

Score: 0 / 6(0 of 6 checked)
  1. 1.

    Solve 5x - 7 = 2x + 8.

    Answer and explanation

    Answer: 5 (also accepted: x = 5, x=5)

    Subtract 2x: 3x - 7 = 8. Add 7: 3x = 15. Divide by 3: x = 5.

  2. 2.

    Solve 4(x + 3) = 2x - 6.

    Answer and explanation

    Answer: -9 (also accepted: x = -9, x=-9)

    Distribute: 4x + 12 = 2x - 6. Subtract 2x: 2x + 12 = -6. Subtract 12: 2x = -18. So x = -9.

  3. 3.

    How many solutions does 3(x - 2) = 3x - 6 have?

    Question 3 options
    Answer and explanation

    Answer: A) Infinitely many

    Distributing gives 3x - 6 = 3x - 6. Subtracting 3x leaves -6 = -6, which is always true, so every number is a solution.

  4. 4.

    Which equation has no solution?

    Question 4 options
    Answer and explanation

    Answer: D) 2x + 5 = 2x - 1

    In 2x + 5 = 2x - 1, subtracting 2x leaves 5 = -1, which is never true, so there is no solution.

  5. 5.

    Solve 0.5(x + 6) = 1.25x - 3.

    Answer and explanation

    Answer: 8 (also accepted: x = 8, x=8)

    Distribute: 0.5x + 3 = 1.25x - 3. Subtract 0.5x: 3 = 0.75x - 3. Add 3: 6 = 0.75x. Divide: x = 8.

  6. 6.

    Solve x/3 + 4 = x - 2.

    Answer and explanation

    Answer: 9 (also accepted: x = 9, x=9)

    Multiply every term by 3: x + 12 = 3x - 6. Subtract x: 12 = 2x - 6. Add 6: 18 = 2x, so x = 9.

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FAQ

How can you tell if an equation has infinitely many solutions?

Simplify both sides fully. If the result is a statement that is always true, such as 4 = 4, every value of the variable works, so there are infinitely many solutions.

Does 8.EE.C.7 include fractions and decimals?

Yes. Part b specifically includes rational number coefficients, so equations with fractions and decimals are expected.

Where does this lead next?

The same skills are needed for systems of equations in 8.EE.C.8 and for the high school standard on solving linear equations and inequalities.

More grade 8 Expressions & Equations standards

8.EE.A.1: Properties of integer exponents8.EE.A.2: Square roots and cube roots8.EE.A.3: Estimating with powers of 108.EE.A.4: Operations in scientific notation8.EE.B.5: Proportional relationships and slope8.EE.B.6: Slope with similar triangles and y = mx + b8.EE.C.8: Systems of linear equations
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