6.03 One-step equations with addition and subtraction

An algebraic equation is a mathematical statement that says two expressions are equal.

There are many ways to represent an algebraic equation. Some of the ways are:

• balance scales

• algebra tiles

• pictorial models

Balance scales are beneficial because they show that the left and right sides of an equation are equal and so the equation is balanced.

The equation shown on this scale is: x-5 = 2

Another way to represent an algebraic equation is to use Algebra tiles. These tiles represent the variables and units on each side of the equation.

We can use the key above, to represent an equation with algebra tiles. The equation x-5 = 2 can be created with this combination of tiles:

Pictorial models can be really helpful for visualizing equations that represent real-world situations. These models can use an image of any object to represent the variables and units of an equation.

Here is a pictorial model of x-5=2 where the variable x is represented by a backpack and the units are represented by a pencil. The gray pencils are meant to show they have been removed from the situation.

Balance scales, algebra tiles, and pictorial models are three different ways that we can better understand and represent algebra equations.

Examples

Example 1

Write the equation represented by the algebra tiles.

Worked Solution

Create a strategy

Apply the idea

There is 1 \, \, +x tile and 4 \,\, +1 tiles on the left side. There are 5 \,\, +1 tiles on the right side. So the equation is:x+4=5

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Example 2

Represent x-3=-5 using algebra tiles. Do not solve the equation.

Worked Solution

Create a strategy

Use a positive variable tile to represent x and -1 tiles to represent the negative constants.

Apply the idea

On the left side of the equation, there is an x which we can represent with one +x tile and a -3 which we can make by using three -1 tiles. On the right side of the equation the -5 can be shown using five -1 tiles.

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Example 3

Write an equation that represents the given balance scale.

Worked Solution

Create a strategy

A balance scale represents an equation that is equal or balanced. The left side of the scale corresponds to the left side of the equation, and the right side of the scale corresponds to the right side of the equation.

Apply the idea

On the left side of the balance there is an x+4 and on the right side of the balance there is an -2. We set the left and right side of the balance to equal:

x+4=-2

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Example 4

Represent x+1=6 using a pictorial model.

Worked Solution

Create a strategy

Choose objects that could represent the variable x and the constants 1 and 6. The objects should be related for it to make sense.

Apply the idea

The coin purse represents x and the coins represent 1 and 6. So the total number of coins in the purse, x, plus 1 coin is equal to 6 total coins.

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When working with equations, we must keep the equation balanced or it will no longer be a true statement.

Adding or subtracting the same amount to both sides keeps the equations balanced. These are two of the properties of equality.

Addition property of equality: adding the same number to both sides of an equation creates an equivalent equation.

Example:\begin{aligned}&\text{If } &x-2 &= 7 \\ &\text{Then } &x-2+2 &= 7+2\end{aligned}

We can also visualize this with a scale.

If

Then

If we add the same amount to both sides of the scale it stays balanced.

Subtraction property of equality: subtracting the same number from both sides of an equation creates an equivalent equation.

Example:\begin{aligned}&\text{If } &x+5 &= 7 \\ &\text{Then } &x+5-5 &= 7-5\end{aligned}

We can also visualize this with a scale.

If

Then

If we take away the same amount from each side of the equation, it will remain balanced.

Notice for these equations the number we chose to add or subtract was the opposite of a number in the original equation. This strategy helps us solve equations by utilizing inverse operations and the additive inverse property.

Addition and subtraction are opposite or inverse operations that undo one another. If we choose the numbers we add or subtract carefully we can use this to eliminate extra numbers from an equation.

For example, with the equation x-5=17 we want to isolate x which requires getting rid of the constant term -5. To do this we can use the inverse operation by adding 5.

x-5+5=17+5

Now we can see the additive inverse in action: x+0=17+5

Now applying the additive identity: x=17+5

Then we can simplify the right side of the equation: x=22

We don't always write out all of these steps, but it is still important to know what is happening algebraically.

Once we have solved an equation, we can verify the solution using the substitution property of equality.

Consider the solution of x=22 for the equation x-5=17.

We can confirm x=22 is the solution by replacing x with 22 in the equation and checking that both sides of the equation are equal.

Since the left and right side of the equation are still equal, we can confirm that x=22 is a solution to the equation.

Examples

Example 5

Scale 1 is a balanced scale.

Scale 1:

Scale 2:

Which of the following options could go in place of the question mark to balance scale 2?

A

B

C

D

Worked Solution

Create a strategy

From Scale 1, we can see that one +1 tile has been added to the left side of the scale. To keep it balanced we must do the same to the right side.

Apply the idea

This will leave a total of three +1 tiles on the right side of the scale, option D.

Reflect and check

We can show this algebraically as:

\displaystyle x	\displaystyle =	\displaystyle 2	Write the equation
\displaystyle x + 1	\displaystyle =	\displaystyle 2 + 1	Add 1 to both sides
\displaystyle x + 1	\displaystyle =	\displaystyle 3	Evaluate

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Example 6

Solve 21 = x + 13

Worked Solution

Create a strategy

The inverse of addition is subtraction, so we need to subtract from both sides of the equation.

Apply the idea

\displaystyle 21	\displaystyle =	\displaystyle x + 13	Original equation
\displaystyle 21-13	\displaystyle =	\displaystyle x+13-13	Subtraction property of equality
\displaystyle 21-13	\displaystyle =	\displaystyle x	Additive inverse
\displaystyle 8	\displaystyle =	\displaystyle x	Evaluate the subtraction

Reflect and check

Let's verify the answer using a balance scale model.

We start with twenty one +1 tiles on the left side of the scale, then one +x and thirteen +1 tiles on the right side to represent 21=x+13.

Based on our solution, 8=x, we can replace the +x tile with 8 of the +1 tiles.

The balance scale is now balanced with both sides having the same number of positive unit tiles \left(+1\right), verifying that our solution is correct.

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Example 7

Solve: x - 1 = 7

Worked Solution

Create a strategy

The inverse of subtraction is addition, so we need to add to both sides of the equation.

Apply the idea

\displaystyle x-1	\displaystyle =	\displaystyle 7	Original equation
\displaystyle x-1+1	\displaystyle =	\displaystyle 7+1	Addition property of equality
\displaystyle x	\displaystyle =	\displaystyle 7+1	Additive inverse
\displaystyle x	\displaystyle =	\displaystyle 8	Evaluate the addition

Reflect and check

Let's verify the answer using a balance scale model.

We start with a +x tile and a -1 tile one the left side of the scale, then 7 of the +1 tiles on the right to represent the equation x-1=7 .

Based on our solution, x=8 we can replace the+x tile with 8 of the +1 tiles.

Remove zero pairs. The scale stays balanced.

The balance scale is now balanced with both sides having the same number of positive unit tiles, verifying that our solution is correct.

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Example 8

-4 is the solution to the equation -8+x=-12.

a

Verify using substitution.

Worked Solution

Create a strategy

Substitute -4 into the equation. The left side of the equation will be equal to the right side if -4 is a solution.

Apply the idea

\displaystyle -8+\left(-4\right)	\displaystyle =	\displaystyle -12	Substitute x=-4
\displaystyle -12	\displaystyle =	\displaystyle -12	Evaluate the addition

The left side of the equation is equal to the right side of the equation, verifying that -4 is a solution.

Reflect and check

We can also use a pictorial model to verify if -4 is the solution to -8+x=-12.

This pictorial model represents the equation -8+x=-12, where each ice cube represents -1 and the cooler represents x.

We then replace the cooler with four ice cubes, representing x=-4.

We now have twelve ice cubes on both sides of the equation, verifying that -4 is a solution to the equation.

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b

Verify using a model.

Worked Solution

Create a strategy

Represent the equation using algebra tiles on a scale. Make the left side of the scale to have the same number of unit tiles as the right side by replacing the variable tile with the appropriate tiles.

Apply the idea

The equation -8+x=-12 can be represented by:

The given solution is -4, so we replace the variable tile with four negative unit tiles.

Both sides of the scale now have twelve negative unit tiles, verifying that -4 is a solution to the equation.

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Example 9

A box of matches contains 500 matches. The match box falls to the ground and you count 78 matches on the ground. The rest of the matches are still in the box.

a

Write an equation that shows the relationship of the number of matches.

Worked Solution

Create a strategy

The amount of matches that are still in the box is the unknown value.

Apply the idea

Let m represent the number of matches still in the box.

m+78=500

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b

Solve the equation and interpret the solution.

Worked Solution

Create a strategy

The inverse of addition is subtraction, so we need to subtract to both sides of the equation.

Apply the idea

\displaystyle m+78	\displaystyle =	\displaystyle 500	Original equation
\displaystyle m+78-78	\displaystyle =	\displaystyle 500-78	Subtraction property of equality
\displaystyle m	\displaystyle =	\displaystyle 422

This means that there are 422 matches that did not fell out of the box.

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Example 10

Write a situation that could represent the equation x+5=25.

Worked Solution

Create a strategy

Think of a situation where there is an unknown value, x added to 5 which would equal 25.

Apply the idea

The amount of money in James' wallet, x, plus the \$ 5 in his hand is equal to the \$ 25 he received for his birthday.

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