In this lesson · jump to a section
Introduction
Interpreting Solutions helps you connect a mathematical rule to a result you can explain and check. Use the graph intersection as an estimate in context. Explain graph resolution and verify the estimated pair against both rules. This lesson belongs to systems of 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 interpreting solutions.
- 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
Solve linear equations and interpret ordered pairs.
Previous lesson: Real-World Applications
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.
Solve two equations together
Use the graph intersection as an estimate in context. Explain graph resolution and verify the estimated pair against both rules.
A solution of a two-variable linear system is an ordered pair satisfying both equations. Graphically it lies at the intersection. Two nonparallel lines have one solution, distinct parallel lines have none, and coincident lines have infinitely many.
Substitution replaces one variable with an equivalent expression from the other equation. Elimination adds or subtracts equivalent equations to cancel a variable; multiplying an equation by a nonzero constant can prepare matching coefficients. Choose a method based on the given structure, then check the final pair in both original equations.
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. Substitute y=12x+19 into the other equation: x+12x+19=123.
- 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. (8,115) satisfies x+y=123; it also satisfies y=12x+19.
- 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 the system y=12x+19 and x+y=123.
- Substitute y=12x+19 into the other equation: x+12x+19=123.
- Combine terms: 13x=104; divide to obtain x=8.
- Substitute back: y=12(8)+19=115. On a graph this ordered pair is the intersection.
Answer: (8, 115)
Verify: (8,115) satisfies x+y=123; it also satisfies y=12x+19.
Avoid this mistake: Checking only one equation.
Example 2 · Basic
Solve the system y=13x+19 and x+y=131.
- Substitute y=13x+19 into the other equation: x+13x+19=131.
- Combine terms: 14x=112; divide to obtain x=8.
- Substitute back: y=13(8)+19=123. On a graph this ordered pair is the intersection.
Answer: (8, 123)
Verify: (8,123) satisfies x+y=131; it also satisfies y=13x+19.
Avoid this mistake: Multiplying only some terms when preparing elimination.
Example 3 · Intermediate
Solve the system y=14x+19 and x+y=139.
- Substitute y=14x+19 into the other equation: x+14x+19=139.
- Combine terms: 15x=120; divide to obtain x=8.
- Substitute back: y=14(8)+19=131. On a graph this ordered pair is the intersection.
Answer: (8, 131)
Verify: (8,131) satisfies x+y=139; it also satisfies y=14x+19.
Avoid this mistake: Reporting one coordinate as the entire solution.
Example 4 · Advanced
44 tickets include adult tickets x and student tickets y. Adult tickets cost $2 and student tickets $1. Revenue is $52. Write and solve the system.
- Counts: x+y=44. Revenue: 2x+y=52.
- Subtract the count equation from the revenue equation: x=8.
- Then y=44−8=36. Both counts are nonnegative whole numbers.
Answer: (8, 36)
Verify: (8,36) satisfies x+y=44; it also satisfies 2x+y=52.
Avoid this mistake: Checking only one equation.
Example 5 · Challenge
Solve the system y=3x+20 and x+y=52.
- Substitute y=3x+20 into the other equation: x+3x+20=52.
- Combine terms: 4x=32; divide to obtain x=8.
- Substitute back: y=3(8)+20=44. On a graph this ordered pair is the intersection.
Answer: (8, 44)
Verify: (8,44) satisfies x+y=52; it also satisfies y=3x+20.
Avoid this mistake: Multiplying only some terms when preparing elimination.
Common mistakes and corrections
- Checking only one equation.
- Multiplying only some terms when preparing elimination.
- Reporting one coordinate as the entire solution.
Real-world applications
- Compare two subscription plans at their break-even point. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Find two ticket quantities from a total count and total revenue. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Use the graph intersection as an estimate in context. Explain graph resolution and verify the estimated pair against both rules.
A solution of a two-variable linear system is an ordered pair satisfying both equations. Graphically it lies at the intersection. Two nonparallel lines have one solution, distinct parallel lines have none, and coincident lines have infinitely many.
Frequently asked questions
What should I know before interpreting solutions?
Solve linear equations and interpret ordered pairs.
How can I check my work?
(8,115) satisfies x+y=123; it also satisfies y=12x+19. 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?
Checking only one equation. Multiplying only some terms when preparing elimination. Reporting one coordinate as the entire solution.
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: Compare two subscription plans at their break-even point.
- 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
- Use the graph intersection as an estimate in context. Explain graph resolution and verify the estimated pair against both rules.
- Checking only one equation.
- Multiplying only some terms when preparing elimination.
- Reporting one coordinate as the entire solution.
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.
Continue learning
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