# 4.02 Modeling balanced equations

Lesson

An equation is a mathematical sentence formed by setting two expressions equal. We say that the two expressions are equal because they have the same value. For example, $6+4=10$6+4=10 because the expression $6+4$6+4 evaluates to $10$10.

We don't need to know the value of every expression in order to write an equation. In fact, we can use algebra to represent equations with unknown values.

### Modeling equations with Algebra tiles

There are many ways to represent an equation. One way is to use Algebra tiles to represent the algebraic expressions on each side.

Positive Negative
Variable tiles  or   or
Unit tiles

Using the key above, let's represent a few equations with Algebra tiles.

#### Worked example

##### Question 1

Model the equation $3x=15$3x=15 with Algebra tiles.

Think: We need to represent the expressions on each side of the equation.

The expression $3x$3x is the same as having the variable three times.

The number $15$15 is the same as having fifteen units.

Do: Place the appropriate Algebra tiles in your workspace. Be sure to separate them by an equal sign.

 $3x$3x $=$= $15$15 The original equation $=$= Three variable tiles Fifteen unit tiles
##### Question 2

Model the equation $x-4=10$x4=10 with Algebra tiles.

Think: We need to represent the expressions on each side of the equation.

The expression $x-4$x4 can also be written as $x+\left(-4\right)$x+(4), so is the same as having one variable tile and four negative unit tiles.

The number $10$10 is the same as having ten positive units.

Do: Place the appropriate Algebra tiles in the workspace. Be sure to separate them by an equal sign.

 $x$x $-4$−4 $=$= $10$10 The original equation $=$= One variable tile Four negative unit tiles Ten positive unit tiles

#### Practice questions

##### Question 3

Write the equation represented by the Algebra tiles. Do not solve the equation.

 $=$=

##### Question 4

Write the equation represented by the Algebra tiles. Do not solve the equation.

 $=$=

## Keeping equations balanced

We want to keep equations balanced so that the two sides of the equals sign remain equivalent. If we don't we could change what the equation means.

Think of a balanced set of scales. The scale remains level when the weights on both sides of the scales are even. The same thing happens with equations.

#### Exploration

This applet represents the equation $x=3$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 $x=3$x=3.

• What sorts of things make the scale imbalanced?
• What sorts of things can you do to keep the scale balanced?
• What are three different equations which are equivalent to $x=3$x=3

### Ways to stay balanced

We can add the same amount to both sides and the equation will stay balanced.

 If we add the same amount to each side of the equation, it will remain balanced.

We can also take away the same amount from both sides.

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

If we double, triple, or even quadruple the amounts on both sides of a scale, the scale will stay balanced. In fact, we can keep it balanced by multiplying or dividing the amounts by any nonzero number - so long as it's the same on both sides!

 If we double the amount on each side, it will remain balanced.

We can also change what amount appears on each side. The same is true for equations. So, $x=3$x=3 is the same equation as $3=x$3=x.

 Items can switch sides.

As we saw with integer addition and subtraction, a zero pair sums to zero. Since it's like adding nothing, it doesn't change the balance either. We can add as many zero pairs as we like and the equation stays the same!

 Adding a zero pair doesn't change the balance.

#### Practice questions

##### Question 5

Scale $1$1 is a balanced scale.

Scale $1$1:

Scale $2$2:

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

A

B

C

D

A

B

C

D

##### Question 6

Scale $1$1 is a balanced scale.

Scale $1$1:

Scale $2$2:

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

A

B

C

D

A

B

C

D

##### Question 7

Scale $1$1 is a balanced scale.

Scale $1$1:

Scale $2$2:

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

A

B

C

D

A

B

C

D

### Outcomes

#### 6.1

Represent relationships between quantities using ratios, and will use appropriate notations, such as a/b , a to b, and a:b

#### 6.13

Solve one-step linear equations in one variable, including practical problems that require the solution of a one-step linear equation in one variable