Two bills tell the rule

The idea

To find y = ax + b from two known pairs of values, substitute each pair to get two equations in a and b, express b from one, substitute into the other, and solve.

Two bills

A telecom company charges a fixed monthly fee plus an amount for every GB of data. A student paid ₹350 for 10 GB and ₹550 for 20 GB. The bill y and the data x follow y = ax + b.

From these two bills, how much do you think each extra GB costs?

• ₹20
• ₹35
• ₹55

Fit both bills

Change a and b until the rule matches both bills.

Solve for a and b

To find y = ax + b from two known pairs of values, substitute each pair to get two equations in a and b. Express b from one, substitute it into the other, and solve.

So the rule is y = 20x + 150.

• Step | Work
• Substitute x = 10, y = 350 | 350 = 10a + b
• Substitute x = 20, y = 550 | 550 = 20a + b
• Express b from the first | b = 350 − 10a
• Put it in the second | 550 = 20a + 350 − 10a
• Solve for a | 10a = 200, so a = 20
• Find b | b = 350 − 200 = 150

Notes

From two pairs of values: substitute each pair into y = ax + b, express b from one equation, substitute it in the other, and solve for a, then b.

Check yourself

A taxi fare follows y = ax + b. For 2 km the fare is ₹40, and for 4 km it is ₹60. Substituting gives 40 = 2a + b and 60 = 4a + b. What is a?

Answer: 10

60 = 4a + 40 − 2a, so 2a = 20 and a = 10.

For the same taxi, what is b? (Use b = 40 − 2a.)

Answer: 20

b = 40 − 2 × 10 = 20. The rule is y = 10x + 20.

What is the first step in finding a and b from two pairs of values?

A rule y = ax + b gives y = 5 when x = 1, and y = 11 when x = 3. What is a?

Answer: 3

11 = 3a + 5 − a, so 2a = 6 and a = 3. Then b = 2.

• Add the two y values. The two pairs must each go into the rule y = ax + b.
• Guess b. Substitute each pair first. Then solve.
• Substitute each pair into y = ax + b to get two equations — correct. Yes. That gives two equations in a and b.