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, draw a diagram.

  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.

$x + 5 = 20$

$x + 20 = 5$

$x = 20 + 5$

$5\boldcdot {20} = x$

$5x=20$

$20x = 5$

 

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, draw 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

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.

Practice Problems ▶

Glossary

solution to an equation

solution to an equation

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

variable

variable

A variable is a letter in an equation.

coefficient

coefficient

In an algebraic expression, the coefficient of a variable is the constant the variable is multiplied by. If the variable appears by itself then it is regarded as being multiplied by 1 and the coefficient is 1.