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Introduction
Linear Relationships helps you connect a mathematical rule to a result you can explain and check. Constant output differences for equal input changes characterize a linear table. Find the rate and initial value separately. This lesson belongs to coordinate geometry and graphing 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 linear relationships.
- 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
Find slope and solve a linear equation.
Previous lesson: Evaluating Functions
Key mathematical vocabulary
- constant
- A value fixed within a problem.
- equation
- A statement that two expressions are equal.
- slope
- Vertical change divided by nonzero horizontal change.
- intercept
- A point where a graph meets an axis.
- domain
- The set of permitted inputs.
Formulas and rules
- y=mx+b
- m=(y₂−y₁)/(x₂−x₁), x₂≠x₁
- y−y₁=m(x−x₁)
Represent a linear function
Constant output differences for equal input changes characterize a linear table. Find the rate and initial value separately.
A nonvertical linear graph has an equation y=mx+b. The slope m describes the change per unit input; b is the output at zero. A table or graph determines a line when it supplies two distinct input-output pairs consistent with a constant rate.
Find m from differences, then use any known point to solve b=y−mx. To graph, plot the y-intercept and use the slope to find a second point; connect them and extend as the domain allows. Point-slope form y−y₁=m(x−x₁) and standard form Ax+By=C are equivalent descriptions after valid rearrangement.
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. The vertical change is 60 and the horizontal change is 5, so m=12.
- 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. The supplied nonzero point satisfies the rule; the slope is unchanged.
- 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
A table contains (0,24), (1,36), and (2,48). Write its linear rule.
- The vertical change is 60 and the horizontal change is 5, so m=12.
- At input zero the output is 24, so the intercept is 24.
- The equation is y = 12x + 24. Preserve equality when changing forms; all forms describe the same points.
Answer: y = 12x + 24
Verify: The supplied nonzero point satisfies the rule; the slope is unchanged.
Avoid this mistake: Using a y-value from any point as the y-intercept.
Example 2 · Basic
A table contains (0,24), (1,37), and (2,50). Write its linear rule.
- The vertical change is 65 and the horizontal change is 5, so m=13.
- At input zero the output is 24, so the intercept is 24.
- The equation is y = 13x + 24. Preserve equality when changing forms; all forms describe the same points.
Answer: y = 13x + 24
Verify: The supplied nonzero point satisfies the rule; the slope is unchanged.
Avoid this mistake: Switching x and y when substituting a point.
Example 3 · Intermediate
A table contains (0,24), (1,38), and (2,52). Write its linear rule.
- The vertical change is 70 and the horizontal change is 5, so m=14.
- At input zero the output is 24, so the intercept is 24.
- The equation is y = 14x + 24. Preserve equality when changing forms; all forms describe the same points.
Answer: y = 14x + 24
Verify: The supplied nonzero point satisfies the rule; the slope is unchanged.
Avoid this mistake: Drawing a line through only one plotted point without using its slope.
Example 4 · Advanced
A table contains (0,25), (1,27), and (2,29). Write its linear rule.
- The vertical change is 10 and the horizontal change is 5, so m=2.
- At input zero the output is 25, so the intercept is 25.
- The equation is y = 2x + 25. Preserve equality when changing forms; all forms describe the same points.
Answer: y = 2x + 25
Verify: The supplied nonzero point satisfies the rule; the slope is unchanged.
Avoid this mistake: Using a y-value from any point as the y-intercept.
Example 5 · Challenge
A table contains (0,25), (1,28), and (2,31). Write its linear rule.
- The vertical change is 15 and the horizontal change is 5, so m=3.
- At input zero the output is 25, so the intercept is 25.
- The equation is y = 3x + 25. Preserve equality when changing forms; all forms describe the same points.
Answer: y = 3x + 25
Verify: The supplied nonzero point satisfies the rule; the slope is unchanged.
Avoid this mistake: Switching x and y when substituting a point.
Common mistakes and corrections
- Using a y-value from any point as the y-intercept.
- Switching x and y when substituting a point.
- Drawing a line through only one plotted point without using its slope.
Real-world applications
- Write a delivery price from a base charge and per-mile cost. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Represent steady filling of a tank with an initial volume. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Constant output differences for equal input changes characterize a linear table. Find the rate and initial value separately.
A nonvertical linear graph has an equation y=mx+b. The slope m describes the change per unit input; b is the output at zero. A table or graph determines a line when it supplies two distinct input-output pairs consistent with a constant rate.
Frequently asked questions
What should I know before linear relationships?
Find slope and solve a linear equation.
How can I check my work?
The supplied nonzero point satisfies the rule; the slope is unchanged. 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?
Using a y-value from any point as the y-intercept. Switching x and y when substituting a point. Drawing a line through only one plotted point without using its slope.
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: Write a delivery price from a base charge and per-mile cost.
- 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
- Constant output differences for equal input changes characterize a linear table. Find the rate and initial value separately.
- Using a y-value from any point as the y-intercept.
- Switching x and y when substituting a point.
- Drawing a line through only one plotted point without using its slope.
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