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.

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.

Write the equation represented by the algebra tiles.

Worked Solution

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

Worked Solution

Write an equation that represents the given balance scale.

Worked Solution

Represent x+1=6 using a pictorial model.

Worked Solution

Idea summary

**Algebraic equation**: mathematical statement that says two expressions are equal.

We can represent algebraic equations pictorially with:

balance scales

algebra tiles

pictorial models

This applet represents the equation x=3.

You can click and drag algebra tiles from the bottom to the gray part at the bottom to be on the scale. Click the reset button in the top right corner to go back to x=3.

Experiment with adding and removing tiles and observe what happens:

What kinds of things can you do that keep the scale balanced?

What kinds of things can you do to make the scale unbalanced?

Test this with different types of tiles. Are these observations always true?

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.

\displaystyle x -5 | \displaystyle = | \displaystyle 17 | Original equation |

\displaystyle \left(22\right)-5 | \displaystyle = | \displaystyle 17 | Substitute 22 for x |

\displaystyle 17 | \displaystyle = | \displaystyle 17 | Evaluate the subtraction |

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

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

Solve 21 = x + 13

Worked Solution

Solve: x - 1 = 7

Worked Solution

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

a

Verify using substitution.

Worked Solution

b

Verify using a model.

Worked Solution

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

b

Solve the equation and interpret the solution.

Worked Solution

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

Worked Solution

Idea summary

Addition property of equality | \text{If } a=b \text{ then } a+ c=b + c |
---|---|

Subtraction property of equality | \text{If } a=b \text{ then } a- c=b -c |

Inverse property of addition | a + \left(-a\right)=0 \text{ and } \left(-a\right) +a=0 |

Identity property of addition | a \cdot 1=a \text{ and } 1 \cdot a = a |

Substitution property | a+0=a \text{ and } 0+a=a |