1. Graphing linear equations and systems of equations
Unit 3
Graphing linear equations and systems of equations

2. Learning target
LT: I can graph proportional relationships. 8.ee.5

3. Proportional relationships
A ratio is a comparison of two numbers.
Example: If there are 13 boys and 12 girls in a class, the ratio of boys to girls is 13 to 12 .
A ratio can be written three ways: 13 to 12, 13:12, and
Note: A ratio is comparing two numbers, so it cannot be written as a mixed number!

4. Proportional relationships
Example 1:
A bouquet contains only red and white roses. The ratio of red roses to white roses is 1:2. What is the ratio of red roses to the total number of roses in the bouquet?
Step 1: write the part-to-part as a fraction.
𝑟𝑒𝑑 𝑟𝑜𝑠𝑒𝑠 𝑤ℎ𝑖𝑡𝑒 𝑟𝑜𝑠𝑒𝑠 = 1 2

5. Proportional relationships
Example 1:
A bouquet contains only red and white roses. The ratio of red roses to white roses is 1:2. What is the ratio of red roses to the total number of roses in the bouquet?
Step 2: Use that ratio to write the part-to-total ratio.
𝑟𝑒𝑑 𝑟𝑜𝑠𝑒𝑠 𝑟𝑒𝑑 𝑟𝑜𝑠𝑒𝑠 + 𝑤ℎ𝑖𝑡𝑒 𝑟𝑜𝑠𝑒𝑠 = 1 3
Solution: The ratio of red roses to total roses is 1:3.

6. Proportional relationships
A proportion shows that two ratios are equal in value.

7. Proportional relationships
Example 2:
The debate team won 3 out of 5 debates it participated in this semester. If the team participated in 20 debates, how many debates did it win? How many did it lose?
Step 1: What does the ratio represent?
3 5 is the ratio of debates won to total debates.

8. Proportional relationships
Example 2:
The debate team won 3 out of 5 debates it participated in this semester. If the team participated in 20 debates, how many debates did it win? How many did it lose?
Step 2: Write a 2nd ratio that includes the same terms.
𝑑𝑒𝑏𝑎𝑡𝑒𝑠 𝑤𝑜𝑛 𝑡𝑜𝑡𝑎𝑙 𝑑𝑒𝑏𝑎𝑡𝑒𝑠 = 𝑥 20

9. Proportional relationships
Example 2:
The debate team won 3 out of 5 debates it participated in this semester. If the team participated in 20 debates, how many debates did it win? How many did it lose?
Step 3: set the ratios equal to each other.
3 5 = 𝑥 20

10. Proportional relationships
Example 2:
The debate team won 3 out of 5 debates it participated in this semester. If the team participated in 20 debates, how many debates did it win? How many did it lose?
Step 3: Use the peanut-walnut method (Cross-Multiplying).
3 5 = 𝑥 20
Solution: The team won 12 debates and lost 8 debates.

11. Proportional relationships
a rate is a ratio that compares quantities that use different units. Use the same strategies to work with rates as you use with other ratios.

12. Proportional relationships
Example 3:
It costs $261 for 3 nights at Pavia Pavilion Hotels. At the same rate, how much will it cost to stay for 7 nights?
Step 1: Write a ratio comparing the cost to the number of nights.
𝑐𝑜𝑠𝑡 # 𝑜𝑓 𝑛𝑖𝑔ℎ𝑡𝑠 =

13. Proportional relationships
Example 3:
It costs $261 for 3 nights at Pavia Pavilion Hotels. At the same rate, how much will it cost to stay for 7 nights?
Step 2: Write a second ratio that includes the unknown
𝑐𝑜𝑠𝑡 # 𝑜𝑓 𝑛𝑖𝑔ℎ𝑡𝑠 = 𝑥 7

14. Proportional relationships
Example 3:
It costs $261 for 3 nights at Pavia Pavilion Hotels. At the same rate, how much will it cost to stay for 7 nights?
Step 3: Set the ratios equal to each other
= 𝑥 7

15. Proportional relationships
Example 3:
It costs $261 for 3 nights at Pavia Pavilion Hotels. At the same rate, how much will it cost to stay for 7 nights?
Step 4: Use Peanut-Peanut Walnut Method.
= 𝑥 7
Solution: a 7-night stay at the pavia pavilion hotels will cost $609.

16. Proportional relationships
A unit rate is a rate that when expressed as a fraction, has a 1 in the denominator.
Ex: 46 miles per gallon is expressed as 46 𝑚𝑖𝑙𝑒𝑠 1 𝑔𝑎𝑙𝑙𝑜𝑛
If a unit rate involves money, it is called a unit price.

17. Proportional relationships
Example 4:
A 16 ounce box of wheaty puffs costs $3.52. a 64 ounce box of wheaty puffs is sold at the same unit price. What is the cost of the 64 ounce box?
Step 1: Find the unit price of the 16 ounce box.
The rate is $ 𝑜𝑢𝑛𝑐𝑒𝑠 . So divide to find the unit price = 3.52÷16 16÷16 =
The unit price is $0.22 per ounce.

18. Proportional relationships
Example 4:
A 16 ounce box of wheaty puffs costs $3.52. a 64 ounce box of wheaty puffs is sold at the same unit price. What is the cost of the 64 ounce box?
Step 2: multiply to find the cost of the 64 ounce box.
$0.22 x 64 = $14.08
Solution: the cost of the 64 ounce box of wheaty puffs is $14.08.

19. Learning target
I can interpret the unit rate of proportional relationships as the slope of the graph. 8.ee.5

20. Direct proportions
A direct proportion is a special kind of linear equation. In a direct proportion, the ratio for two variables, such as x and y, is a constant m. that means that for every change in x, y changes by a constant factor, m.
A direct proportion may be written in the following forms:
Y =mx or 𝑦 𝑥 = m where m ≠0 and m is the constant of proportionality, as well as the slope of the line.

21. Direct proportions Example 1: see document camera.
Step 1: find the ratio 𝑦 𝑥 for the two points on the line in graph 1.
For (-1, 6) the ratio is 6 −1 = -6.
For (4, -4) the ratio is −4 4 = -1.
The ratios are not constant and are not a direct proportion.

22. Direct proportions Example 1: see document camera.
Step 2: find the ratio 𝑦 𝑥 for the two points on the line in graph 2.
For (-3, 1) the ratio is 1 −3 =
For (6, -2) the ratio is −2 6 =
The ratio is constant for both points on the line. This graph shows a direct proportion.

23. Direct proportions
Example 2: During a car trip, a car is driven at a constant rate of 60 miles per hour on the highway.
Write an equation and make a graph to display the total distance that the car will travel if it maintains that speed for at least three hours. What does the slope of the graph represent?

24. Direct proportions
Example 2: During a car trip, a car is driven at a constant rate of 60 miles per hour on the highway.
Step 1: Will the equation you write be a direct proportion?
The car travels at a constant rate, so the total distance is directly proportional to the number of hours the car is driven.

25. Direct proportions
Example 2: During a car trip, a car is driven at a constant rate of 60 miles per hour on the highway.
Step 2: Write an equation in the form y=mx.
Let x represent the number of hours.
Let y represent the total distance, in miles.
The car travels at a constant rate of 60 𝑚𝑖𝑙𝑒𝑠 1 ℎ𝑜𝑢𝑟 .
The equation is y=60x.

26. Direct proportions
Example 2: During a car trip, a car is driven at a constant rate of 60 miles per hour on the highway.
Step 3: draw and label a coordinate grid.
Only use quadrant 1 because a car cannot travel a negative distance.
Title the graph and label its axes.

27. Direct proportions
Example 2: During a car trip, a car is driven at a constant rate of 60 miles per hour on the highway.
Step 4: find two points and connect them with a line.
Plot the 1st point at (o,0). Since it’s a direct proportion it must pass through the origin.
Find another point by substituting a value for x. I picked 3 hours.
Y = 60(3)
Y = 180
Plot the 2nd point at (3, 180). Connect the points.

28. Direct proportions
Example 2: During a car trip, a car is driven at a constant rate of 60 miles per hour on the highway.
Step 5: What does the slope of the graph represent?
In y=60x, m=60. so, the slope of the graph, 60, shows the speed of the car.