# Lesson 2

Truth and Equations

Let's use equations to represent stories and see what it means to solve equations.

### 2.1: Three Letters

1. The equation $$a + b = c$$ could be true or false.

1. If $$a$$ is 3, $$b$$ is 4, and $$c$$ is 5, is the equation true or false?
2. Find new values of $$a$$, $$b$$, and $$c$$ that make the equation true.
3. Find new values of $$a$$, $$b$$, and $$c$$ that make the equation false.
2. The equation $$x \boldcdot y = z$$ could be true or false.

1. If $$x$$ is 3, $$y$$ is 4, and $$z$$ is 12, is the equation true or false?
2. Find new values of $$x$$, $$y$$, and $$z$$ that make the equation true.
3. Find new values of $$x$$, $$y$$, and $$z$$ that make the equation false.

### 2.2: Storytime

Here are three situations and six equations. Which equation best represents each situation? If you get stuck, consider drawing a diagram.

$$x + 5 = 20$$

$$x + 20 = 5$$

$$x = 20 + 5$$

$$5\boldcdot {20} = x$$

$$5x=20$$

$$20x = 5$$

1. After Elena ran 5 miles on Friday, she had run a total of 20 miles for the week. She ran $$x$$ miles before Friday.

2. Andre’s school has 20 clubs, which is five times as many as his cousin’s school. His cousin’s school has $$x$$ clubs.

3. Jada volunteers at the animal shelter. She divided 5 cups of cat food equally to feed 20 cats. Each cat received $$x$$ cups of food.

### 2.3: Using Structure to Find Solutions

Here are some equations that contain a variable and a list of values. Think about what each equation means and find a solution in the list of values. If you get stuck, consider drawing a diagram. Be prepared to explain why your solution is correct.

1. $$1000 - a = 400$$
2. $$12.6 = b + 4.1$$
3. $$8c = 8$$
4. $$\frac23 \boldcdot d = \frac{10}{9}$$
5. $$10e = 1$$
6. $$10 = 0.5f$$
7. $$0.99 = 1 - g$$
8. $$h + \frac 3 7 = 1$$

List:

$$\frac18$$

$$\frac37$$

$$\frac47$$

$$\frac35$$

$$\frac53$$

$$\frac73$$

0.01

0.1

0.5

1

2

8.5

9.5

16.7

20

400

600

1400

One solution to the equation $$a+b+c=10$$ is $$a=2$$, $$b=5$$, $$c=3$$

How many different whole-number solutions are there to the equation $$a+b+c=10$$? Explain or show your reasoning.

### Summary

An equation can be true or false. An example of a true equation is $$7+1=4 \boldcdot 2$$. An example of a false equation is $$7+1=9$$

An equation can have a letter in it, for example, $$u+1=8$$. This equation is false if $$u$$ is 3, because $$3+1$$ does not equal 8. This equation is true if $$u$$ is 7, because $$7+1=8$$.

A letter in an equation is called a variable. In $$u+1=8$$, the variable is $$u$$. A number that can be used in place of the variable that makes the equation true is called a solution to the equation. In $$u+1=8$$, the solution is 7.

When a number is written next to a variable, the number and the variable are being multiplied. For example, $$7x=21$$ means the same thing as $$7 \boldcdot x = 21$$. A number written next to a variable is called a coefficient. If no coefficient is written, the coefficient is 1. For example, in the equation $$p+3=5$$, the coefficient of $$p$$ is 1.

### Glossary Entries

• coefficient

A coefficient is a number that is multiplied by a variable.

For example, in the expression $$3x+5$$, the coefficient of $$x$$ is 3. In the expression $$y+5$$, the coefficient of $$y$$ is 1, because $$y=1 \boldcdot y$$.

• solution to an equation

A solution to an equation is a number that can be used in place of the variable to make the equation true.

For example, 7 is the solution to the equation $$m+1=8$$, because it is true that $$7+1=8$$. The solution to $$m+1=8$$ is not 9, because $$9+1 \ne 8$$

• variable

A variable is a letter that represents a number. You can choose different numbers for the value of the variable.

For example, in the expression $$10-x$$, the variable is $$x$$. If the value of $$x$$ is 3, then $$10-x=7$$, because $$10-3=7$$. If the value of $$x$$ is 6, then $$10-x=4$$, because $$10-6=4$$.