Unknown numbers

Explore the following equations and determine the unknown numbers. How can you balance each equation?

Recall that the same symbol or letter means the same number.

$\mathrm{♥}\mathrm{ }+\mathrm{ }\mathrm{♥}\mathrm{ }+\mathrm{ }\mathrm{♥} = 12$

$b + b + b = 24$

$s + s + s + s + s + s = 48$

$s + s + s + s + s = 45$

What strategies did you use? Share your thoughts with a peer, if possible.

What's an equation?

By figuring out the value of the letters and symbols in the Minds On section, you were solving equations!

An equation is a mathematical statement that has equivalent expressions on both sides of an equal sign.

Notice and name

There are a variety of ways (or strategies) to solve equations and find out what the value of variables could be.

Explore the following strategies that can be used to solve equations.

Let's consider the example of the balance scale that was provided in the image.

Inspection

$♥+♥+♥=12\text{or}3\times ♥=12$

By recalling the 'times 3' facts and by inspecting or looking at this equation, we know we need the number that when multiplied by 3 will equal 12. That number is 4.

Depending upon the size of the numbers in the equation, building a model may be a starting point.

Consider $b+b+b=24.$

First, simplify the equation. Group like terms (all the bs) $3b=24.$

Next, use 24 counters.

It is 3 groups, but how many are in a group?

One method is to start making three groups with counters.

Number them off 1, 2, 3 and consider how many need to have in each group.

Did you find 8?

Next, check your work, substituting 8 for the variable.

$\begin{array}{rl}3b & = 24\\ 3\left(8\right) & = 24\\ 24 & = 24, b = 8\end{array}$

Guess and check

$s + s + s + s + s + s = 48$

Simplify first! Group like terms.

$6s = 48$

You now know that $5\times 6$ is 30, so $s$ must be bigger than that, but not much bigger because 30 isn’t that far away from 48, so you can try 7.

Substitute:

$6\left(7\right)=42$ too small

$6\left(8\right)=48, \mathrm{s}=8.$

You don’t need to verify your solution as you’ve already “checked” it when you substituted your guess in.

Inverse/opposite operation

$s + s + s + s + s = 45$

Simplify first! Group like terms.

$5s=45,$ the opposite of multiply is divide, so you divide 45 by 5, giving you 9.

Let’s check:

$\begin{array}{rl}5\left(9\right) & = 45\\ 45 & = 45,s = 45\end{array}$

Balance method

This works alongside opposite operations.

$\begin{array}{rl}3s + 5 & = 14\\ 3s + 5 – 5 & = 14 – 5\\ 3s & = 9\\ \frac{3s}{3}& = \frac{9}{3}\\ s & = 3\end{array}$

Time to check:

$\begin{array}{rl}3 \left(3\right) + 5 & = 14\\ 9 + 5 & = 14\\ 14 & = 14,s = 3\end{array}$

Another way to solve equations with more than one operation like $\frac{\mathit{s}}{4}-2=10$  is to use a reverse flow chart.

And then, check your result.

$\begin{array}{rl}\frac{48}{4} -2 & = 10\\ 12 – 2 & = 10\\ 10 & = 10\\ s & = 48\end{array}$

Your turn

Different strategies work for different situations.

Solve the following algebraic equations and check your solutions using any or all of the strategies described above.

a) $3s = 21$

b) $4b = 24$

c) $3x – 5 = 7$

d) $45 + 54 = 11t$

Record your work using a method of your choice, or by using the following fillable and printable document, Algebraic Equations.

After you have tried solving the equations, you may press ‘Answers’ to reveal the answers to check your work.

Answers

Writing and solving equations

An artist saved each week for 5 weeks, and then spent $25 on new paints.

How much did the artist save every week if after the 5 weeks, and after having spent $25, there’s still $100 left?

Write an equation that reflects the situation, solve it, and check your result.

Reflect on the strategies for solving equations.

Which strategy do you use most often? Why?

Record your answers using a method of your choice.

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.

Now, record your ideas using a voice recorder, speech-to-text, or writing tool.