1. Equations
Lesson 2.2.1

2. Equations 2.2.1 California Standard: What it means for you: Key Words:
Lesson
2.2.1
Equations
California Standard:
Algebra and Functions 1.1
Write and solve one-step linear equations in one variable.
What it means for you:
You’ll learn about equations and how they’re related to expressions.
Key Words:
equation
guess and check
solve

3. Lesson
2.2.1
Equations
You’ve done plenty of work with expressions — now it’s time to tackle equations.
They’re quite similar — both involve variables, numbers, and combinations of the usual +, –, ×, and ÷ operations.

4. Equations 2.2.1 An Equation Tells You That Two Things are Equal
Lesson
2.2.1
Equations
An Equation Tells You That Two Things are Equal
An equation tells you that something is equal to something else.
These are all examples of equations:
9 – 3 = 4 + 2
= 20
4y + x = x – 7
Writing an equation is similar to writing an expression. It may involve a real-life problem, and you may have to use variables to represent unknown values.

5. Lesson
2.2.1
Equations
Example 1
Richard spent $3.50 on fruit and $7.26 on vegetables. He then spent $x on meat, making a total of exactly $20. Write an equation to represent this.
Solution
Richard spent x dollars, which is equal to $20.
Writing this as an equation, x = 20.
This can be simplified to 10.76 + x = 20.
Solution follows…

6. Lesson
2.2.1
Equations
Example 2
Edward and Bianca each wrote a story in English class. Bianca’s story has 31 words more than Edward’s. Write an equation to represent this.
Solution
Create variables — call the number of words in Bianca’s story b and the number of words in Edward’s story e.
Although both b and e are unknown amounts, the question says that b is 31 greater than e.
So, a possible equation is e + 31 = b. You could also have written b – 31 = e.
Both equations show that b is 31 greater than e.
Solution follows…

7. Equations 2.2.1 Guided Practice
Lesson
2.2.1
Equations
Guided Practice
Write equations to represent the situations described
in Exercises 1–3.
1. Jose scored 10 percent more than William on a class test.
2. Theresa owns two pairs of shoes more than Kimberly.
3. Richard and Tim each bake cakes. Richard notices that Tim’s cake is three times as heavy as his.
Using j for Jose’s percent, and w for William’s: either j – 10 = w or w + 10 = j
Using t for the number of pairs of shoes owned by Theresa, and k for the number owned by Kimberly: either t – 2 = k or k + 2 = t
Using r for the weight of Richard’s cake and t for Tim’s: t = 3r
Solution follows…

8. Equations 2.2.1 Guided Practice
Lesson
2.2.1
Equations
Guided Practice
Write equations to represent the situations described
in Exercises 4–6.
4. Isaac goes to the county fair. Because he has been saving up, he has seven times as much money as his sister to spend on rides.
5. Laura’s dad runs a newsstand that sells bottles of water. Laura knows that five small bottles of water contain the same amount as two large bottles.
6. Monica has a candy bar that is a third the size of Julio’s.
Using i for Isaac’s money and s for his sister’s: i = 7s
Using s for the volume of a small bottle and b for a big: 5s = 2b
Using m for the size of Monica’s bar and j for the size of Julio’s: m = j ÷ 3
Solution follows…

9. Equations 2.2.1 Equations Can Be Used to Find the Value of Variables
Lesson
2.2.1
Equations
Equations Can Be Used to Find the Value of Variables
Equations like y + 6 = 16 tell you enough information to find out y.
One way to find y is to guess at it again and again until you find the right answer.
Sometimes this is called the “guess and check” method.

10. Equations 2.2.1 Find the value of y if y + 6 = 16. Solution
Lesson
2.2.1
Equations
Example 3
Find the value of y if y + 6 = 16.
Solution
First you need to guess a value for y. Then you can use it in the equation and check if it makes the equation true.
Try y = 5: = 16 is not true. In fact, = 11, which is too small.
Try y = 12: = 16 is also not true. In fact, = 18, which is too big.
Try y = 10: = 16, which is true. So y does equal This means the solution to the equation is y = 10.
Solution follows…

11. Lesson
2.2.1
Equations
Example 4
Julia uses the expression 7h to find out how many dollars she earned doing yard work. h represents the number of hours she worked. If Julia earned $56, then how many hours did she work?
Solution
An equation for the situation is 7h = 56.
Try guessing different values of h, then see if they’re right. h 7h
7h = 56? 7 49 No 10 70 8 56
Yes
If Julia worked for 7 hours, she would have earned only $49, which is too little.
If Julia worked for 10 hours, she would have earned $70, which is too much.
Julia must have worked for 8 hours — because when h = 8, the equation 7h = 56 is true.
Solution follows…

12. Equations 2.2.1 Guided Practice
Lesson
2.2.1
Equations
Guided Practice
Use “guess and check” to solve the equations
in Exercises 7–10.
7. x + 14 = 30
8. 2z + 16 = 38
9. 9a – 12 = 69
10. 3z ÷ 7 = 6
x = 16
z = 11
a = 9
z = 14
Solution follows…

13. Equations 2.2.1 Guided Practice
Lesson
2.2.1
Equations
Guided Practice
11. Luis uses the expression $3 × m to work out how much money he spends calling his relatives in Europe for m minutes.
His last phone call cost $60.
Write an equation to represent this situation and then find out how long he spoke for.
3m = 60, Luis spoke for 20 minutes
Solution follows…

14. Equations 2.2.1 Guided Practice
Lesson
2.2.1
Equations
Guided Practice
12. Pedro works out how much he must pay toward a food bill by using the formula d – $3.24, where d is the cost of the entire bill.
He ends up paying $2.97.
Write an equation for this situation and then find out how much the food bill must have been.
13. What value of y satisfies the equation 111 = 19y + 16?
d – 3.24 = 2.97, the food bill was $6.21
E.g. Try y = 10, 111 = 19 × = = 206 – too big
Try y = 5, 111 = 19 × = = 111, so y = 5.
Solution follows…

15. Equations 2.2.1 Independent Practice
Lesson
2.2.1
Equations
Independent Practice
Use “guess and check” to solve the equations
in Exercises 1–4.
1. a + 20 = 57
2. 3b + 9 = 15
3. 7c – 22 = 20
4. 12d ÷ 8 = 6
a = 37
b = 2
c = 6
d = 4
Solution follows…

16. Equations 2.2.1 Independent Practice
Lesson
2.2.1
Equations
Independent Practice
5. Brandon has $t in his savings jar. He puts in an extra $12.
If the total amount in the jar is n, write an equation linking n and t.
6. Solve the equation you wrote for Exercise 5 when n = 48.
7. Cody is playing an old video game. In it you get 400 points every time you collect a ring and 800 points every time you collect an egg.
Write an equation about p, the total number of points you get for collecting r rings and e eggs.
t + 12 = n
t = 36
p = 400r + 800e
Solution follows…

17. Equations 2.2.1 Independent Practice
Lesson
2.2.1
Equations
Independent Practice
8. Guadalupe has started going to dance class.
Each class costs $3 and she spent $23.50 on dance shoes. Including the shoes, she has spent $71.50 in total.
Write an equation for this and solve it to find out how many lessons Guadalupe has been to.
9. It takes 2 eggs to make 20 cookies, and 4 eggs to make 40 cookies.
Which equation best describes the number of eggs, x, that it takes to make y cookies?
If n is the number of lessons Guadalupe has been to, = n.
Guadalupe has been to 16 lessons.
y = 10x
x = 10y
y + x = 10
y – x = 10
Solution follows…

18. Lesson
2.2.1
Equations
Round Up
Equations look like expressions, but they contain a “=” — which tells you that something is equal to something else.
Don’t worry if you’re not 100% comfortable with equations yet — all of the Lessons in this Section are about equations, so you’ll have many chances to practice.