Åk 6–9

4.1 From Problems to Algebra
When solving problems, it is often effective to be able to set up equations.  When you use equations, you make it easier if you have well thought-out strategy.

Imagine that you have the following problem in front of you:
 Hot dog eating contest There is a hotdog eating contest with three participants.  In total they ate 44 hotdogs.  Urban ate 3 more hotdogs than Olivia and Viktor hate for less than Olivia. How many hotdogs did each of them eat and who won the contest? Bild: Hogia AB, Multimediabyrån

Try to guess your way to the answer and see if you can find how many hotdogs each of them ate.

If you don’t succeed, you can solve the problem with and equation.  Here comes a suggestion on how you can work it out.

1. Ask yourself the questions – what are we looking for and what do I know?

Well – you know how many hotdogs they ate together and how the numbers relate to each other but not the number of hotdogs that each of them ate which is what is asked for.

2. Write down an expression for how many hotdogs each of them ate.

You can choose to write mathematically the number of hotdogs with for one of them. We choose Olivia because the other two are compared to her.

We suppose that Olivia ate x hotdogs which is written as x.

Urban ate 3 more than Olivia.
Urban ate x hotdogs + 3 hotdogs which is written as x + 3

Victor ate for less than Olivia.
Victor ate x hotdogs - 4 hotdogs which is written as x - 4 3. Set up the equation. You know that they ate 44 hotdogs together.

Urban’s hotdogs + Olivia’s hotdogs + Viktor’s hotdogs = 44 hotdogs

x + 3 + x + x – 4 = 44

4. Solve the equation

x + 3 + x + x – 4 = 44

3x - 1 = 44

3x = 45

x = 15

5. Check that the solution is correct

The equation: x + 3 + x + x – 4 = 44

Check with x = 15;

15 + 3 + 15 + 15 – 4 = 44

LHS = RHS, it is correct.

We now go back to the different expressions and replace x with 15.

Olivia = x hotdogs = 15 hotdogs
Urban = x hotdogs + 3 hotdogs = 15 + 3 = 18 hotdogs
Viktor = x hotdogs – 4 hotdogs = 15 – 4 = 11 hotdogs

Answer: Olivia ate 15 hotdogs, Urban ate 18 hotdogs and Viktor ate 11 hotdogs.

Urban then won the hotdog eating contest.

Problem Solving by using Figures and Equations
A very good way to simplify equation solving is to use figures.

Example:
Tyra cycles twice as far as Linnea and Dejan cycles 5 km farther than Linnea. In total they cycle 45 km together. How far do they each cycle?

Draw a figure and then it becomes much easier to write an equation.

We assume that Linnea cycles x km
Tyra cycles: 2x km
Dejan cycles r: x + 5 km

Together they have cycled 45 km. We set up an equation: x + 2x + x + 5 = 45

Solve the equation:
x + 2x + x + 5 = 40

4x = 40

x = 10

We check that it is correct:
10 + 2 · 10 + 10 + 5 = 45

It is correct!