English:Linear Relationships
Linear Relationships
Introduction
This course is designed for learners in Grades 7–8. A linear relationship connects two quantities so that equal changes in one quantity produce equal changes in the other. When you graph the relationship on a coordinate plane, its points lie on a straight line. Linear relationships are useful for describing situations such as a taxi fare with a starting fee, money earned at a constant hourly rate, distance traveled at a steady speed, or the temperature change in a simple model.
In this aiMOOC, you will learn to recognize linear relationships in words, tables, graphs, and equations. You will also learn to calculate and interpret slope, identify the y-intercept, write equations in the form y = mx + b, compare different linear relationships, and decide when a real-world situation can be modeled by a line.
The coordinate plane gives you a common place to connect tables, graphs, and equations. The horizontal axis is the x-axis, the vertical axis is the y-axis, and their intersection is the origin. A point is written as an ordered pair such as (3, 5), where the first value tells you the horizontal position and the second tells you the vertical position.
Learning Goals
By the end of this course, you should be able to explain what makes a relationship linear, calculate a constant rate of change, find and interpret slope and y-intercept, move between tables, graphs, equations, and verbal descriptions, distinguish proportional from non-proportional linear relationships, and use linear models to solve problems.
Understanding Linear Relationships
Variables and Relationships
A variable represents a quantity that can change. In many school examples, x is the input or independent variable and y is the output or dependent variable. The value of y depends on the value chosen for x.
Suppose you earn $8 for each hour of a small job. If x is the number of hours and y is the amount earned, then y changes by $8 whenever x increases by 1. The relationship can be written as y = 8x. The constant change of $8 per hour is the rate of change.
A relationship is linear when its rate of change stays constant over the interval being considered. In a table, equal changes in x produce equal changes in y. In a graph, the points form a straight line. In an equation, a nonvertical linear relationship can be written in a form such as y = mx + b.
Four Ways to Represent a Linear Relationship
You can represent the same linear relationship in several ways.
- Verbal description: A sentence explains how one quantity changes with another, such as “A bike rental costs $6 to start and $4 for each hour.”
- Table of values: Ordered pairs show matching x- and y-values.
- Graph: Points are plotted on a coordinate plane and form a straight line.
- Equation: A rule such as y = 4x + 6 summarizes the relationship.
Being able to move between these representations is an important algebra skill. Each representation emphasizes something different: a table shows numerical patterns, a graph shows shape and direction, an equation gives a compact rule, and a verbal description gives meaning and units.
Rate of Change and Slope
Constant Rate of Change
The rate of change tells you how much the output changes for a given change in the input. For a linear relationship, this rate is constant.
If a table has the points (1, 7), (2, 10), (3, 13), and (4, 16), then each increase of 1 in x produces an increase of 3 in y. The constant rate of change is 3 units of y for every 1 unit of x.
If x changes by more than 1, compare the changes using the ratio:
rate of change = change in y ÷ change in x
For example, if x increases from 2 to 6 while y increases from 5 to 17, then the change in x is 4 and the change in y is 12. The rate of change is 12 ÷ 4 = 3.
Slope
The slope of a line is its rate of change. It is usually represented by the letter m.
m = change in y ÷ change in x
You may also hear this described as rise over run. The rise is the vertical change and the run is the horizontal change.
A positive slope means the line rises from left to right. A negative slope means the line falls from left to right. A slope of zero produces a horizontal line. A vertical line has an undefined slope because its horizontal change is zero, so the slope ratio would require division by zero.
Finding Slope from Two Points
To find slope from two points, subtract the y-values in the same order that you subtract the x-values.
Suppose a line passes through (2, 5) and (6, 13).
The change in y is 13 - 5 = 8. The change in x is 6 - 2 = 4. Therefore, the slope is 8 ÷ 4 = 2.
You would get the same result if you reversed both subtraction orders: 5 - 13 = -8 and 2 - 6 = -4, so -8 ÷ -4 = 2. What matters is using the same point order in both differences.
The y-Intercept and Slope-Intercept Form
The y-Intercept
The y-intercept is the y-value where a nonvertical line crosses the y-axis. At that point, x = 0. In a real-world problem, the y-intercept often represents a starting amount, an initial fee, or a value that already exists before the input begins to increase.
For example, a music club charges a $12 registration fee plus $5 for each month. If x is the number of months and y is the total cost, then the starting amount is $12. The y-intercept is 12.
The Equation y = mx + b
A common way to write a linear equation is:
y = mx + b
Here, m is the slope and b is the y-intercept.
For the equation y = 3x + 4, the slope is 3 and the y-intercept is 4. That means the graph crosses the y-axis at 4 and rises 3 units for every 1 unit moved to the right.
Writing an Equation from a Situation
Suppose a streaming service charges a one-time setup fee of $10 and then $7 per month. Let x represent the number of months and y represent the total cost.
The monthly charge is the rate of change, so m = 7. The setup fee is the starting value, so b = 10. The equation is:
y = 7x + 10
Check the meaning of the equation. At x = 0, the cost is $10. Each time x increases by 1 month, y increases by $7.
Graphing Linear Relationships
Graphing from an Equation
To graph y = 2x + 1, start with the y-intercept. Plot the point (0, 1). Then use the slope 2, which can be read as 2 over 1. Move up 2 and right 1 to reach another point such as (1, 3). Repeat the same movement to find more points, then draw a straight line through them.
You can also make a table. Choose x-values, substitute them into the equation, and plot the resulting ordered pairs.
Reading a Graph
When you read a graph, ask four questions. What quantities are on the axes? What units are used? Where does the line cross the y-axis? How quickly does the line rise or fall?
A graph should be interpreted in context. A slope of 5 could mean 5 dollars per hour, 5 kilometers per minute, or 5 degrees per day. The number alone is not enough; the units tell you what the slope means.
Proportional and Non-Proportional Linear Relationships
Proportional Relationships
A proportional relationship has the form y = mx. Its y-intercept is 0, so its graph passes through the origin. The constant of proportionality is also the slope.
For example, if apples cost $3 per kilogram with no fixed fee, the cost can be modeled by y = 3x. Buying 0 kilograms costs $0, so the graph includes the origin.
Non-Proportional Linear Relationships
A linear relationship can still be linear even when it is not proportional. If the equation has a nonzero y-intercept, it has the form y = mx + b with b not equal to 0.
For example, a taxi fare might be modeled by y = 2x + 5, where x is distance and y is cost. The slope 2 represents the price per unit of distance, while the y-intercept 5 represents the starting fare. The graph is a straight line but does not pass through the origin.
This distinction is important: all proportional relationships are linear, but not all linear relationships are proportional.
Recognizing Linear and Nonlinear Patterns
From a Table
A table represents a linear relationship when equal changes in x produce equal changes in y.
Consider these y-values for x = 0, 1, 2, 3: 4, 7, 10, 13. The y-values increase by 3 each time, so the rate of change is constant and the relationship is linear.
Now consider y-values 1, 4, 9, 16 for x = 1, 2, 3, 4. The changes in y are 3, 5, and 7. Because the change is not constant, this pattern is not linear.
From a Graph
A linear graph is straight. If the graph curves, bends, or changes its steepness, the rate of change is not constant over the whole graph.
Two different straight lines can have different slopes and y-intercepts. Their steepness tells you how their rates of change compare, while their y-intercepts tell you how their starting values compare.
From an Equation
Equations such as y = 4x - 3, y = -2x + 7, and y = 5 are linear. The variable x is to the first power and is multiplied only by a constant.
Equations such as y = x², y = 2^x, or y = 1/x are not linear. Their graphs do not have one constant slope across their domains.
Comparing Linear Relationships
Comparing Slopes
A larger positive slope means a faster increase. Among y = 2x + 1 and y = 5x - 4, the second relationship increases more quickly because its slope is 5 rather than 2.
For negative slopes, the size and sign both matter. A slope of -4 means y decreases by 4 when x increases by 1, while a slope of -1 means y decreases by only 1 for the same increase in x.
Comparing Starting Values
The y-intercept lets you compare initial values. If Plan A has y = 3x + 20 and Plan B has y = 4x + 10, then Plan A starts higher because 20 is greater than 10, but Plan B grows faster because its slope is 4 rather than 3.
This kind of comparison helps you reason about when one plan may become more expensive than another. The point where two lines meet represents an input value at which the two relationships have the same output.
Building Linear Models from Real Situations
Choosing Variables and Units
Before writing an equation, decide what x and y represent. Then write down their units. For example, x could represent hours and y could represent dollars. The slope would then have units of dollars per hour, and the y-intercept would have units of dollars.
Units help you check whether your model makes sense. If a problem is about distance over time, a slope measured in kilometers per hour has a clear meaning. A slope written without units can hide mistakes.
A Modeling Example
A water tank already contains 40 liters and fills at 6 liters per minute. Let x be the number of minutes after filling begins and y be the number of liters in the tank.
The initial amount is 40, so b = 40. The rate is 6 liters per minute, so m = 6. The model is y = 6x + 40.
After 5 minutes, y = 6 × 5 + 40 = 70, so the model predicts 70 liters.
A useful model also has a reasonable domain. Negative time would not fit this situation, and the model should stop being used after the tank reaches its capacity.
When a Linear Model Is Only an Approximation
Real situations are not always perfectly linear. A car may not travel at exactly the same speed, prices may change, and a tank may stop filling when it is full. A linear model is useful when the rate of change is approximately constant over the interval you care about.
Good mathematical modeling means asking both, “Does the equation fit the data?” and “Does the equation make sense in the real situation?”
Common Mistakes and How to Avoid Them
Mixing Up Slope and y-Intercept
In y = mx + b, the coefficient multiplying x is the slope, while the separate constant is the y-intercept. For y = -3x + 8, the slope is -3 and the y-intercept is 8.
Forgetting the Sign of a Negative Slope
If a line falls from left to right, its slope is negative. Keep track of whether the change in y is positive or negative.
Using Unequal x-Steps Without Adjusting
If x increases by 2 instead of 1, do not compare only the y-change. Divide the change in y by the change in x.
Assuming Every Straight-Line Situation Is Proportional
A straight-line relationship with a nonzero y-intercept is linear but not proportional. To be proportional, the graph must pass through the origin.
Interactive Tasks
Quiz: Test Your Knowledge
Which statement best describes a linear relationship? (It has a constant rate of change) (!Its output must always be positive) (!Its graph must always pass through the origin) (!Its slope must always be greater than one)
What does the slope of a line represent? (The rate of change) (!The largest y-value) (!The location of the origin) (!The number of plotted points)
In the equation y = 4x + 7, what is the slope? (four) (!seven) (!eleven) (!zero)
In the equation y = 4x + 7, what is the y-intercept? (seven) (!four) (!eleven) (!negative seven)
Which rule represents a proportional relationship? (y equals five times x) (!y equals five times x plus two) (!y equals x times x) (!y equals five plus x times x)
A line rises 12 units while moving 3 units to the right. What is its slope? (four) (!nine) (!fifteen) (!thirty six)
What does a negative slope tell you? (The line falls from left to right) (!The line is vertical) (!The line crosses the origin) (!The y-intercept is negative)
Which table pattern is evidence of a linear relationship when x increases by one each time? (The y-values change by the same amount each time) (!The y-values are all different) (!The x-values are all positive) (!The y-values become larger)
A gym charges a starting fee of 20 dollars and 6 dollars per visit. Which rule models the total cost y after x visits? (y equals six times x plus twenty) (!y equals twenty times x plus six) (!y equals twenty six times x) (!y equals six times x)
What must be true for a linear relationship to be proportional? (Its y-intercept is zero) (!Its slope is zero) (!Its y-intercept is positive) (!Its slope is negative)
Memory Game
| Slope | Constant rate of change of a line |
| Intercept | Value where a line crosses an axis |
| Origin | Point where the coordinate axes meet |
| Proportionality | Relationship whose graph passes through zero zero |
| Variable | Quantity that can change |
| Equation | Mathematical statement showing that two expressions are equal |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Positive slope | Line rises from left to right |
| Negative slope | Line falls from left to right |
| Zero slope | Horizontal line |
| Y-intercept | Starting output when the input is zero |
| Constant rate of change | Equal input changes produce equal output changes |
...
Crossword Puzzle
| Slope | What word names the constant rate of change of a line? |
| Intercept | What word names the value where a line crosses an axis? |
| Coordinate | What word names one number used to locate a point on an axis? |
| Proportional | What kind of linear relationship passes through the origin? |
| Equation | What word names a mathematical statement that shows two expressions are equal? |
| Constant | What word describes a value or rate that does not change? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Linear pattern hunt: Find three everyday situations that might have a constant rate of change, name the two variables in each situation, and explain what you think the rate means.
- Table maker: Create a five-row table for y = 3x + 2, describe the numerical pattern, and explain how the table shows the slope and y-intercept.
- Graph sketch: Draw a coordinate plane and sketch one line with positive slope, one with negative slope, and one with zero slope; label each line and describe its direction.
- Slope story: Write a short real-world story that could be modeled by y = 4x + 10 and explain what 4 and 10 mean in your story.
Standard
- Representation poster: Make a poster that shows one linear relationship as a verbal description, table, graph, and equation, with arrows explaining how the representations connect.
- Mini interview: Interview an adult about a situation involving a starting cost plus a regular rate, record the quantities involved, and decide whether a linear model is reasonable.
- Data collection: Measure a quantity at equal time intervals during a safe classroom or home activity, graph the data, and judge whether the pattern is approximately linear.
- Plan comparison: Invent two pricing plans with different slopes and y-intercepts, graph both equations, and explain which plan is better for small and large input values.
Advanced
- Linear modeling video: Produce a two- to three-minute teaching video that explains how to build y = mx + b from a real-world situation and includes one worked example.
- Error analysis: Create three believable mistakes students might make when finding slope or y-intercept, then write corrections that explain why each mistake is wrong.
- Piecewise investigation: Find or invent a situation that is linear for one interval but changes rate later, represent it with two line segments, and explain why one equation is not enough for the whole situation.
- Model evaluation: Collect or locate a small data set that is nearly linear, create a line that models it, discuss prediction error, and explain where using the model would become unreasonable.
Learning Assessment
- Representation transfer: Given a verbal situation with a fixed starting value and a constant rate, create a table, graph, and equation, then explain how the same slope and y-intercept appear in all three representations.
- Model comparison: Compare two linear models with different starting values and rates, determine which output is larger for several inputs, and justify how the slopes and intercepts influence your conclusion.
- Reasoning from data: Analyze an unfamiliar table with uneven x-steps, decide whether it is linear by calculating rates of change, and explain your reasoning.
- Context interpretation: Interpret the slope and y-intercept of a linear equation in a real context, including correct units and a statement about whether each value is realistic.
- Model critique: Evaluate a proposed linear model for a real situation, identify a useful domain, and explain at least one reason the model could fail outside that domain.
- Nonlinear contrast: Compare a linear pattern with a nonlinear pattern and explain how their tables and graphs reveal the difference in rate of change.
Evidence of Learning
- Knowledge: You can explain linear relationships, slope, rate of change, y-intercept, proportionality, and the meaning of y = mx + b.
- Skills: You can calculate slope, identify intercepts, test tables for constant rate of change, graph lines, write equations, and move between verbal, tabular, graphical, and algebraic representations.
- Products: You can create accurate tables, graphs, equations, posters, written explanations, data investigations, and short teaching media about linear models.
- Reasoning: You can justify why a relationship is or is not linear, compare two models, interpret units, check whether results are reasonable, and explain common errors.
- Transfer: You can recognize and build linear models in unfamiliar situations such as pricing, motion, measurement, saving, or resource use, while identifying where a model should and should not be applied.
OERs on the Topic
The following English Wikipedia article gives a broader reference for linear functions. At this level, focus especially on the straight-line graph, constant rate of change, slope, and intercept ideas.
Linked Learning Areas