# Introduction to Simultaneous Equations

Videos to help Grade 8 students learn that that a system of linear equations, also known as simultaneous equations, is when two or more equations are involved in the same problem and work must be completed on them simultaneously.

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Lesson Plans and Worksheets for Grade 8

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## New York State Common Core Math Module 4, Grade 8, Lesson 24

Lesson 24 Student Outcomes

• Students know that a system of linear equations, also known as simultaneous equations, is when two or more equations are involved in the same problem and work must be completed on them simultaneously. Students also learn the notation for simultaneous equations.
• Students compare the graphs that comprise a system of linear equations, in the context of constant rates, to answer questions about time and distance.

Lesson 24 Student Summary

Simultaneous linear equations, or a system of linear equations, is when two or more linear equations are involved in the same problem. Simultaneous linear equations are graphed on the same coordinate plane.
The solution to a system of linear equations is the set of all points that make the equations of the system true. If given two equations in the system, the solution(s) must make both equations true.

Lesson 24 Opening Exercise

Opening Exercises 1–3
1. Derek scored 30 points in the basketball game he played and not once did he go to the free throw line. That means that Derek scored two point shots and three point shots. List as many combinations of two and three pointers as you can that would total points.
Write an equation to describe the data.

2. Derek tells you that the number of two-point shots that he made is five more than the number of three-point shots. How many combinations can you come up with that fit this scenario? (Don’t worry about the total number of points.)
Write an equation to describe the data.

3. Which pair of numbers from your table in Exercise 2 would show Derek’s actual score of points?

Example 1
Pia types at a constant rate of pages every minutes. Suppose she types pages in minutes. Pia’s constant rate can be expressed as the linear equation y = 1/5 x.
Pia typically begins work at 8:00 a.m. every day. On our graph, her start time is reflected as the origin of the graph that is, zero minutes worked and zero pages typed. For some reason, she started working 5 minutes earlier today. How can we reflect her extra time working today on our graph?
Does a translation of units to the right reflect her working an additional minutes?
Does a translation of units to the left reflect her working an additional minutes?
What is the equation that represents the graph of the translated line?
If Pia started work 20 minutes early, what equation would represent the number of pages she could type in minutes?

Example 2
Now we will look at an example of a situation that requires simultaneous linear equations. Sandy and Charlie walk at constant speeds. Sandy walks from their school to the train station in 15 minutes and Charlie walks the same distance in 10 minutes. Charlie starts 4 minutes after Sandy left the school. Can Charlie catch up to Sandy? The distance between the school and the station is miles.
What is Sandy’s average speed in 15 minutes? Explain.
What is Charlie’s average speed in 10 minutes? Explain.

Example 3
Randi and Craig ride their bikes at constant speeds. It takes Randi 25 minutes to bike 4 miles. Craig can bike 4 miles in 32 minutes. If Randi gives Craig a 20 minute head start, about how long will it take Randi to catch up to Craig?

Exercises
4. Efrain and Fernie are on a road trip. Each of them drives at a constant speed. Efrain is a safe driver and travels 45 miles per hour for the entire trip. Fernie is not such a safe driver. He drives 70 miles per hour throughout the trip. Fernie and Efrain left from the same location, but Efrain left at 8:00 a.m. and Fernie left at 11:00 a.m. Assuming they take the same route, will Fernie ever catch up to Efrain? If so, approximately when?
a. Write the linear equation that represents Efrain's constant speed. Make sure to include in your equation the extra time that Efrain was able to travel.
b. Write the linear equation that represents Fernie's constant speed.
c. Write the system of linear equations that represents this situation.
d. Sketch the graph.
e. Will Fernie ever catch up to Efrain? If so, approximately when?
f. At approximately what point do the graphs of the lines intersect?

5. Jessica and Karl run at constant speeds. Jessica can run 3 miles in 15 minutes. Karl can run 2 miles in 8 minutes. They decide to race each other. As soon as the race begins, Karl realizes that he did not tie his shoes properly and takes 1 minute to fix them.
a. Write the linear equation that represents Jessica's constant speed. Make sure to include in your equation the extra time that Jessica was able to run.
b. Write the linear equation that represents Karl's constant speed.
c. Write the system of linear equations that represents this situation.
d. Sketch the graph.
e. Use the graph to answer the questions below.
i. If Jessica and Karl raced for 2 miles. Who would win? Explain.
ii. If the winner of the race was the person who got to a distance of 1/2 mile first, who would the winner be? Explain.
iii. At approximately what point would Jessica and Karl be tied? Explain.