Minds On
Balance the equation
Investigate numbers that will make this equation balanced and true. Record your thinking in a format of your choice. If possible, discuss your thinking with a partner.
- Did you have to guess the answers for each shape until you found combinations that made the equation true?
- How many combinations did you find?
Action
Guess and check
We will examine a strategy called guess and check to solve algebraic equations. Let’s begin by accessing the video “Solving Equations Using Guess and Check” in which Ms. Liu will explain how to solve equations using this strategy!
Examine the following word problem:
Three cars are driving to a city from the same starting point. Car 2 is 2 kilometres farther than Car 1, and Car 3 is 4 kilometres farther than Car 2. Together they have travelled 22 kilometres. Find the distance travelled by each car. There is a table with different distances travelled by each car to predict and figure out what distance each car needs to travel for a total distance of 20 kilometres. On the side of the table, there is a tip to start with the shortest (smallest) distance. The first prediction or Option A is car 1 travels 6 kilometres, car 2 travels 8 kilometres and car 3 travels 12 kilometres. This makes 26 kilometres in total which is too high. The second prediction or Option B is car 1 travels 65kilometres, car 2 travels 7 kilometres and car 3 travels 11 kilometres. This makes 23 kilometres in total which is also too high. The third prediction or Option C is car 1 travels 4 kilometres, car 2 travels 6 kilometres and car 3 travels 10 kilometres. This makes exactly 20 kilometres which is the correct answer.
Student Tips
Student tips
Notice how we use the information to make a logical guess. We then add it up to check and see if the total is 20. Sometimes it may take more than one guess to get the right solution.
Your turn
A painter has 24 paintbrushes. There are 6 more used paintbrushes than clean paintbrushes. How many clean and used paintbrushes are there?
Press the ‘Answer’ button to check your work.
clean + used = 24
used = clean + 6 (6 more used than clean. If clean is 1, then used is 7, and so on).
(clean + 6) + clean = 24
(2 × clean) + 6 = 24
2 × clean = 24 − 6
2 × clean = 18
The final answer is: clean = 9 and used = 15
Like terms in equations
In algebraic equations, a letter next to a number signifies multiplication. 5T means 5 × T.
You can add and subtract variables that are the same, like how you would with whole numbers.
This is called collecting like terms. Like terms are terms whose variables are the same.
For example, in the equation for paintbrushes, you had clean + used but then you substituted (clean + 6) for used.
Let the letter c represent the word clean.
You ended up with clean + clean + 6 = 24 or c + c + 6 = 24
If you had an equation like this one:
c +c + 6 = 24
You could add both of the terms of the equation together because they are both “c ”s
2c + 6 = 24
What would be the value of the variable c?
2 times c plus 6 is equal to 24. You can use a guess and check method to find a value for c that will make the equation true.
Let’s try the number 3.
(2 × 3) + 6 = 12, this is not the correct value for c because 12 does not equal 24.
(2 × 5) + 6 = 16, this is not the correct value for c because 16 does not equal 24.
(2 × 9) + 6 = 24, this is the correct value for c because 24 is equal to 24.
When we guess and check, we examine the equation, and using our knowledge of numbers we take a logical guess as to what the best solution might be, and then we check to see if we are correct.
Let’s practice using the guess and check strategy with the following questions.
3a + 4a = 21
7a = 21 (Combine like terms)
We can use a guess and check strategy to solve this.
We can start by selecting a number for a. Let’s select the number 2. Then, try it out:
7(2) = 14
14 ≠ 21
Which means the value 2 when multiplied by 7 does not produce the product we are looking for.
We can then try a larger value. For example, the number 3:
7(3) = 21
21 = 21
Therefore, the value of 3 for a is correct.
We can also check our answer by substituting our value for a into the equation:
3a + 4a = 21
a= 3
3(3) + 4(3)
= 9 + 12
= 21
Our guess was correct a = 3.
Test Your Skills!
Test your skills
Complete the following fillable and printable document Guess & Check Worksheet. You can also complete this activity in your notebook or using a method of your choice.
Press the ‘Activity’ button to access Guess & Check Worksheet.
Consolidation
Solve the equations
Complete the following two questions. Use a method of your choice to record your thinking and solutions.
- There were 14 fans at a basketball game. One team had two more fans than the other team. How many fans did both teams have?
- Create a similar question of your own. Find the solution. There must be at least two clues.
Reflection
As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.
I feel...
Now, record your ideas using a voice recorder, speech-to-text, or writing tool.