The cab costs something before it even moves. Then it charges per mile.
A cab charges $3 before the ride starts, then $2 for every mile. The graph is a straight line, so this is a linear relationship. We can write y = mx + b. In this example, x is miles traveled and y is cost in dollars. The slope m tells how much y changes when x increases by 1. The y-intercept b is y when x = 0. Here m = 2 and b = 3. Other problems can use x and y for different quantities.
Sit down and the meter already says $3. Then $2 every mile. m = 2, b = 3, so y = 2x + 3. After 4 miles you pay 2 × 4 + 3 = $11.
If b is 0, there is no start fee. The line goes through (0, 0), and it is also proportional: y is always the same number times x. If b is not 0, it is still a straight line. It is just not proportional.
In a table, divide the change in y by the matching change in x to find slope m. If x increases by 1, m is simply the change in y. On a graph, this is rise divided by run: vertical change divided by horizontal change. A downward change is negative. The y-intercept b is the y-value where the line crosses the y-axis, at x = 0.
Try it. Then open the steps.
Example 1 of 5
A ride costs $3 plus $2 per mile. Write y = mx + b, then find the cost of 7 miles.
Answer: y = 2x + 3, $17
Example 2 of 5
A line goes through (0, 5) and (4, 17). Write the equation.
Answer: y = 3x + 5
Example 3 of 5
Saturday, a leak under the sink
A plumber charges $75 to show up and $40 per hour. Cost of a 3-hour job?
Answer: $195
Example 4 of 5
Sign-up sheet
A gym is $20 to join, then $8 per class. Cost of 6 classes?
Answer: $68
Example 5 of 5
Checking the family phone bill
A plan is $15 plus $0.12 per text. A bill is $21. How many texts?
Answer: 50 texts
Easy to miss: Calling every linear graph proportional.
Do this: Ask: does it go through (0, 0)? If not, it is linear but not proportional.
Easy to miss: Mixing up m and b. Using the starting fee as the slope.
Do this: m is the repeating change. b is the one-time start.
26 problems · Camp, Trail, Summit
