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Introduction
Equations with Decimals helps you connect a mathematical rule to a result you can explain and check. Multiply the whole equation by a power of ten to obtain integer coefficients before solving. Never change only the decimal terms. This lesson belongs to linear equations and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in equations with decimals.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Use distribution, signed arithmetic, and inverse operations.
Previous lesson: Equations with Fractions
Key mathematical vocabulary
- variable
- A letter representing an unknown or changing quantity.
- coefficient
- A factor multiplying a variable term.
- constant
- A value fixed within a problem.
- expression
- A mathematical phrase representing a value.
- equation
- A statement that two expressions are equal.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Preserve equality while solving
Multiply the whole equation by a power of ten to obtain integer coefficients before solving. Never change only the decimal terms.
An equation states that two expressions have the same value. A solution is an input that makes the original statement true. Adding, subtracting, or multiplying both sides by the same number preserves equality; division is permitted only by a nonzero number.
First remove grouping symbols and combine like terms if needed. Gather variable terms on one side and constants on the other, then isolate the variable. An equation can have one solution, no solution, or infinitely many solutions. A false constant statement means no solution; a true identity means all permitted inputs work.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Multiply every term by 10: 12x+29=65.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Substitute x=3 into the ORIGINAL equation: both sides equal 6.5.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Solve 1.2x + 2.9 = 6.5. Explain each inverse operation.
- Multiply every term by 10: 12x+29=65.
- Subtract 29: 12x=36. Divide by 12: x=3.
- Verification: Substitute x=3 into the ORIGINAL equation: both sides equal 6.5.
Answer: 3
Verify: Substitute x=3 into the ORIGINAL equation: both sides equal 6.5.
Avoid this mistake: Changing only one side: apply each operation to both sides.
Example 2 · Basic
Solve 1.3x + 2.9 = 6.8. Explain each inverse operation.
- Multiply every term by 10: 13x+29=68.
- Subtract 29: 13x=39. Divide by 13: x=3.
- Verification: Substitute x=3 into the ORIGINAL equation: both sides equal 6.8.
Answer: 3
Verify: Substitute x=3 into the ORIGINAL equation: both sides equal 6.8.
Avoid this mistake: Dropping a negative coefficient during division.
Example 3 · Intermediate
Solve 1.4x + 2.9 = 7.1. Explain each inverse operation.
- Multiply every term by 10: 14x+29=71.
- Subtract 29: 14x=42. Divide by 14: x=3.
- Verification: Substitute x=3 into the ORIGINAL equation: both sides equal 7.1.
Answer: 3
Verify: Substitute x=3 into the ORIGINAL equation: both sides equal 7.1.
Avoid this mistake: Assuming every equation has one numerical solution: inspect identities and contradictions.
Example 4 · Advanced
A rental costs $2 per hour plus $30. The total is $36. Write an equation and find the hours x.
- Model cost: 2x+30=36.
- Subtract the fixed fee 30: 2x=6.
- Divide by the hourly rate 2: x=3 hours.
Answer: 3
Verify: Substitute x=3 into the ORIGINAL equation: both sides equal 36.
Avoid this mistake: Changing only one side: apply each operation to both sides.
Example 5 · Challenge
Solve 0.3x + 3 = 3.9. Explain each inverse operation.
- Multiply every term by 10: 3x+30=39.
- Subtract 30: 3x=9. Divide by 3: x=3.
- Verification: Substitute x=3 into the ORIGINAL equation: both sides equal 3.9.
Answer: 3
Verify: Substitute x=3 into the ORIGINAL equation: both sides equal 3.9.
Avoid this mistake: Dropping a negative coefficient during division.
Common mistakes and corrections
- Changing only one side: apply each operation to both sides.
- Dropping a negative coefficient during division.
- Assuming every equation has one numerical solution: inspect identities and contradictions.
Real-world applications
- Find hours worked from a fixed fee and hourly charge. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Determine the quantity of supplies affordable at a known total cost. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Multiply the whole equation by a power of ten to obtain integer coefficients before solving. Never change only the decimal terms.
An equation states that two expressions have the same value. A solution is an input that makes the original statement true. Adding, subtracting, or multiplying both sides by the same number preserves equality; division is permitted only by a nonzero number.
Frequently asked questions
What should I know before equations with decimals?
Use distribution, signed arithmetic, and inverse operations.
How can I check my work?
Substitute x=3 into the ORIGINAL equation: both sides equal 6.5. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Changing only one side: apply each operation to both sides. Dropping a negative coefficient during division. Assuming every equation has one numerical solution: inspect identities and contradictions.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Find hours worked from a fixed fee and hourly charge.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Multiply the whole equation by a power of ten to obtain integer coefficients before solving. Never change only the decimal terms.
- Changing only one side: apply each operation to both sides.
- Dropping a negative coefficient during division.
- Assuming every equation has one numerical solution: inspect identities and contradictions.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
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Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials