Introduction to Linear Polynomials · Lesson 5 of 7
Linear Relationships
“Two changing quantities finally define the relationship instead of saying, “It’s complicated.””
• Understand the general form y = ax + b for a linear relationship. • Interpret a and b in a real context. • Use two data points to determine a linear relationship. • Solve the pair of equations that arises from two observations. • Apply the method to cost and temperature relationships.
A linear relationship describes how two variables are connected when one of them changes at a constant rate compared with the other. This means that whenever the input changes by the same amount, the output also changes by the same fixed amount. Because this rate of change stays constant, the relationship can be represented by a straight line on a graph.
The general form of a linear relationship is (y = ax + b). Here, (x) represents the input or independent variable, while (y) represents the corresponding output or dependent variable.
The number (a) tells us how quickly (y) changes when (x) changes, so it represents the constant rate of change. The number (b) gives the value of (y) when (x = 0). Together, (a) and (b) determine the position and direction of the straight line.
From a Pattern to a Relationship
For the square-tile pattern, if x is the stage number and y is the number of tiles, then y = 2x − 1. The same rule that generated the sequence now becomes a relationship between two variables.
Finding a and b from Two Observations
If we know two points on a linear relationship, we can substitute both into y = ax + b. This gives two equations in the two unknowns a and b. Solving them reveals the complete relationship.
Problem
A bill is ₹350 for 10 GB of data and ₹550 for 20 GB. Assume y = ax + b. Find a and b.
- 1.For x = 10, y = 350: 350 = 10a + b.
- 2.For x = 20, y = 550: 550 = 20a + b.
- 3.Subtract the first equation from the second: 200 = 10a.
- 4.So a = 20.
- 5.Substitute into 350 = 10a + b: 350 = 200 + b.
- 6.Therefore b = 150.
- 7.The relationship is y = 20x + 150.
| Quantity | Meaning in y = 20x + 150 |
|---|---|
| 20 | Additional cost per GB |
| 150 | Fixed monthly charge |
| x | Data used in GB |
| y | Total bill |
A Faster View of the Same Calculation
The change in y is 550 − 350 = 200 while the change in x is 20 − 10 = 10. Therefore y changes by 20 for each increase of 1 in x. That is exactly a = 20. Once a is known, b can be found by substituting either data point.
This rate of change will later be called the slope of the line. In this chapter, it is enough to understand it as the constant change in y for each unit change in x.
Example with a Learning Platform
Problem
A platform bill is ₹400 for 10 modules and ₹500 for 14 modules. Find y = ax + b.
- 1.Use points (10,400) and (14,500).
- 2.a = (500 − 400)/(14 − 10) = 100/4 = 25.
- 3.Substitute into 400 = 25(10) + b.
- 4.400 = 250 + b, so b = 150.
- 5.Therefore y = 25x + 150.
Example with Court Usage
Problem
A bill is ₹800 for 10 hours and ₹1100 for 15 hours. Find y = ax + b.
- 1.a = (1100 − 800)/(15 − 10) = 300/5 = 60.
- 2.Substitute into 800 = 60(10) + b.
- 3.800 = 600 + b, so b = 200.
- 4.Therefore y = 60x + 200.
Temperature Conversion as a Linear Relationship
The same method works even when the variables are not money and usage. Suppose Celsius temperature C and Fahrenheit temperature F satisfy C = aF + b. We know that ice melts at (F,C) = (32,0) and water boils at (212,100). These two points determine the relationship.
Problem
Find C = aF + b using (32,0) and (212,100).
- 1.a = (100 − 0)/(212 − 32) = 100/180 = 5/9.
- 2.Use 0 = (5/9)(32) + b.
- 3.b = −160/9.
- 4.Therefore C = (5/9)F − 160/9.
- 5.Equivalently, C = (5/9)(F − 32).
Practice Problems
- A platform charges ₹400 for 10 modules and ₹500 for 14 modules. Find y = ax + b.
- A gym charges ₹800 for 10 hours and ₹1100 for 15 hours. Find the linear relationship between cost and hours.
- Using the melting and boiling points of water, derive the Celsius-Fahrenheit relationship in the form C = aF + b.
- For the tile pattern y = 2x − 1, explain what 2 and −1 represent in terms of the rule.
Key Takeaways
• A linear relationship between two variables can be written as y = ax + b. • Two distinct data points determine a and b. • a is the constant rate of change. • b is the value of y when x = 0. • The same method applies to cost, usage, temperature and many other contexts.
Next, we turn linear relationships into straight-line graphs and study the roles of slope and y-intercept visually.