Problem Solving

Linear equations can be applied to many real world applications. They are useful to solve practical problems we face everyday. First, we need to identify if a scenario is linear by checking if the change in variables is constant. Here are a couple examples:

  • A car drives 60 km/h. Every hour, the distance of the car changes by 60 km.
  • The cost of a soccer ball is $29.99. The total cost changes by $29.99 for every ball you purchase.
  • A tub is filled with water at a rate of 5 L/s. The volume of the tub increases by 5 L every second.

If you are given a table of values, you can always check that the first difference is constant meaning the relationship is linear.

After you confirm the scenario is linear, we will need to write a linear equation in the form:

\( y = mx + b\)


  • \(y\) is the dependent variable
  • \(x\) is the independent variable
  • \(m\) is the slope or rate of change
  • \(b\) is the y-int or starting point (when \(x = 0 \))

It is a good idea to use varibles that match the scenario (for example, \(t\) for time). It is also good practice to write out what each variable represents and also the units. For example, 't represents the time in hours'.

The cost of a rental car is $22 plus $12 per day. Write an equation to model the relationship.

The cost of a rental car is $22 plus $12 per day. Calcualte the cost of a one week car rental.

Throughout the week starting Monday, there were 12 sodas in Mei's fridge. Mei would drink them everyday and by Tuesday there were only 10 sodas left and only 8 on Wednesday. This pattern countinued until Mei drank all the remaining of sodas. Create an equation and a graph to show the relationship between the the amount of sodas left each day.