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Finding Linear Relationships

Formulate a linear relationship y = ax + b from given conditions.

7 lessons

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content

The Form y = ax + b

Introduce finding relationships using y = ax + b.

A linear relationship between two variables $x$ and $y$ can often be expressed as the equation $y = ax + b$ . In this equation, $a$ and $b$ are constants that determine the specific rule connecting the variables.

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Solving for Constants

If you have two pairs of

(x, y)

values, you can substitute them into

y = a x + b

to create two separate equations. This gives you a system of linear equations that allows you to solve for the constants

a

and

b

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Telecom Bill Calculation

Worked example solving for a and b.

Problem. A telecom company charges a fixed monthly fee and an additional cost per GB of internet data used. A student observes that when she used 10 GB, her bill was ₹350. When she used 20 GB, her bill was ₹550. If the monthly bill $y$ depends on the amount of data used, $x$ (in GB), according to the relation $y = ax + b$ , find the values of $a$ and $b$ .

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Worksheet

Gym Charges

Faded example based on Exercise 2.5 Q2.

0 of 3 blanks filled

Let's formulate the linear relationship $y = ax + b$ for the gym charges, where $b$ is the fixed monthly fee and $a$ is the hourly cost. For 10 hours of use, the bill is ₹800, giving the equation: $800 =$ $a + b$ .

For 15 hours, the bill is ₹1100, which gives the equation: $1100 = 15a + b$ . From the first equation, we can express the fixed fee as $b = 800 - 10a$ .

Substituting this expression into the second equation yields: $1100 = 15a + (800 - 10a)$ , which simplifies to $300 =$ $a$ .

Solving this gives the hourly rate $a =$ , and substituting back gives the fixed fee $b = 200$ .

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sequence

Order the Solving Steps

Order the steps to find a and b.

Drag the steps into the correct logical order for solving simultaneous linear equations to find constants 'a' and 'b'.

Drag to reorder

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feedback

Temperature Conversion

Guided problem based on Exercise 2.5 Q3.

Consider the relationship between temperature in degrees Celsius (°C) and Fahrenheit (°F) given by

C = aF + b

. Given that ice melts at

0^\circ C

and

32^\circ F

, and water boils at

100^\circ C

and

212^\circ F

, solve for the constants

a

and

b

step-by-step.

Identify the given (F, C) pairs

List the two pairs of coordinate values provided in the problem.

Write the two equations

Substitute the given pairs into C = aF + b.

Isolate 'b' in the first equation

Rearrange your simpler equation to express b in terms of a.

Substitute and solve for 'a'

Substitute your expression for b into the second equation and solve for a.

Solve for 'b' and state the final values

Calculate b using the value of a, then state both constants.

content

The Form y = ax + b

Introduce finding relationships using y = ax + b.

1 / 3

content

Telecom Bill Calculation

Worked example solving for a and b.

1 / 6

Worksheet

Gym Charges

Faded example based on Exercise 2.5 Q2.

0 of 3 blanks filled

For 15 hours, the bill is ₹1100, which gives the equation: $1100 = 15a + b$ . From the first equation, we can express the fixed fee as $b = 800 - 10a$ .

Substituting this expression into the second equation yields: $1100 = 15a + (800 - 10a)$ , which simplifies to $300 =$ $a$ .

Solving this gives the hourly rate $a =$ , and substituting back gives the fixed fee $b = 200$ .

feedback

Temperature Conversion

Guided problem based on Exercise 2.5 Q3.

Consider the relationship between temperature in degrees Celsius (°C) and Fahrenheit (°F) given by

C = aF + b

. Given that ice melts at

0^\circ C

and

32^\circ F

, and water boils at

100^\circ C

and

212^\circ F

, solve for the constants

a

and

b

step-by-step.

Identify the given (F, C) pairs

List the two pairs of coordinate values provided in the problem.

Write the two equations

Substitute the given pairs into C = aF + b.

Isolate 'b' in the first equation

Rearrange your simpler equation to express b in terms of a.

Substitute and solve for 'a'

Substitute your expression for b into the second equation and solve for a.

Solve for 'b' and state the final values

Calculate b using the value of a, then state both constants.