# Systems of equations

Contents

A "system of equations" is when we're dealing with more than one equation at the same time. These tutorials show you how to set up and solve systems of equations.

## Intro to systems of equations

Whether in the real world or a cliche fantasy one, systems of equations are key to solving super-important issues like "the make-up of change in a troll's pocket". This tutorial shows you how to solve systems of equations graphically.

6:51

Systems of equations: trolls, tolls (1 of 2)

A troll forces us to use algebra to figure out the make-up of his currency. We end up setting up a system of equations.

5:41

Systems of equations: trolls, tolls (2 of 2)

Solving the system of equations visually. Now we can save the prince/princess.

2:38

Testing a solution to a system of equations

Sal checks whether (-1,7) is a solution of the system: x+2y=13 and 3x-y=-11.

Exercise

Test solutions to systems of equations

Verify whether a given pair of values is a solution to a system of equations.

## Systems of equations with graphing

We already know that we can represent the set of all x-y pairs that satisfy a linear equation as a line. If there is a point where two of these lines intersect, then the x-y pair corresponding to that point must satisfy both equations.
Common Core Standards: 8.EE.C.8, 8.EE.C.8a

Article

Systems of equations with graphing

Walk through examples of solving systems of equations by finding the point of intersection.

8:30

Systems of equations with graphing

Sal shows how to solve a system of linear equations by graphing and looking for the point of intersection.

6:35

Systems of equations with graphing: 5x+3y=7 & 3x-2y=8

Sal solves the system of equations 5x + 3y = 7 and 3x - 2y = 8 by graphing.

2:22

Systems of equations with graphing: y=7/5x-5 & y=3/5x-1

Sal graphs the following system of equations and solves it by looking for the intersection point: y=7/5x-5 and y=3/5x-1.

5:19

Systems of equations with graphing: chores

Graphical Systems Application Problem

Exercise

Systems of equations with graphing

Graph a system of equations in slope-intercept or standard form, and find their solution using the graphs.

## Solving systems with elimination

You can solve a system of equations with either substitution or elimination. This tutorial focuses with a ton of examples on elimination. You'll learn best if you pause the videos and try to do the problem before Sal does. Once you get a hang for things, feel free to skip forward to the exercises.
Common Core Standards: 8.EE.C.8, 8.EE.C.8b

Article

Systems of equations with elimination

Walk through examples of solving systems of equations with elimination.

3:45

Systems of equations with elimination: 3t+4g=6 & -6t+g=6

Sal solves the system of equations 3t + 4g = 6 and -6t + g = 6 using elimination.

3:38

Systems of equations with elimination: x+2y=6 & 4x-2y=14

Sal solves the system of equations x + 2y = 6 and 4x - 2y = 14 using elimination.

4:26

Systems of equations with elimination: -3y+4x=11 & y+2x=13

Sal solves the system of equations -3y + 4x = 11 and y + 2x = 13 using elimination.

5:05

Systems of equations with elimination: 2x-y=14 & -6x+3y=-42

Sal solves the system of equations 2x - y = 14 and -6x + 3y = -42 using elimination.

4:57

Systems of equations with elimination: 4x-2y=5 & 2x-y=2.5

Sal solves the system of equations 4x - 2y = 5 and 2x - y = 2.5 using elimination.

4:18

Systems of equations with elimination: x-4y=-18 & -x+3y=11

Sal solves the following system of equations by eliminating x: x-4y=-18 and -x+3y=11.

Exercise

Systems of equations with elimination

Solve systems of equation with one-step elimination (e.g., x-values or y-values cancel each other out).

5:31

Systems of equations with elimination: 6x-6y=-24 & -5x-5y=-60

Sal solves the system of equations 6x - 6y = -24 and -5x - 5y = -60 using elimination.

Exercise

Systems of equations with elimination challenge

Solve systems of equation with multi-step elimination (e.g., a manipulation is needed in order for x-values or y-values to cancel each other out).

## Solving systems with substitution

This tutorial is focused on solving systems through substitution. It has more examples than you can shake a dog at. As always, pause the video and try to solve before Sal does. Once you get a hang for things, feel free to skip the rest of the videos and try the exercises. The best way to learn, after all, is to do rather than just listen!
Common Core Standards: 8.EE.C.8, 8.EE.C.8b

Article

Systems of equations with substitution

Walk through examples of solving systems of equations with substitution.

4:39

Systems of equations with substitution: 2y=x+7 & x=y-4

Sal solves the system of equations 2y = x + 7 and x = y - 4 using substitution.

4:06

Systems of equations with substitution: y=4x-17.5 & y+2x=6.5

Sal solves the system of equations y = 4x - 17.5 and y + 2x = 6.5 using substitution.

4:21

Systems of equations with substitution: -3x-4y=-2 & y=2x-5

Sal solves the system of equations -3x - 4y = -2 and y = 2x - 5 using substitution.

6:04

Systems of equations with substitution: 9x+3y=15 & y-x=5

Sal solves the system of equations 9x + 3y = 15 and y - x = 5 using substitution.

Exercise

Systems of equations with substitution

Solve systems of equations where one of the equations is solved for one of the variables.

3:44

Systems of equations with substitution: y=-5x+8 & 10x+2y=-2

Sal solves the system of equations y = -5x + 8 and 10x + 2y = -2 using substitution.

3:44

Systems of equations with substitution: y=-1/4x+100 & y=-1/4+120

Sal solves the system of equations y = -1/4x + 100 and y = -1/4 + 120 using substitution.

## Systems of equations word problems

This tutorial doesn't involve talking parrots and greedy trolls, but it takes many of the ideas you might have learned in that tutorial and applies them to word problems. These include rate problems, mixture problems, and others. If you can pause and solve the example videos before Sal does, we'd say that you have a pretty good grasp of systems. Enjoy!
Common Core Standards: 8.EE.C.8, 8.EE.C.8b, 8.EE.C.8c

Exercise

Solve linear systems of equations

Solve systems of equations with any number of solutions using any solution method.

5:06

Systems of equations with elimination: apples and oranges

Sal solves a word problem about the price of apples and oranges by creating a system of equations and solving it.

5:51

Systems of equations with elimination: TV & DVD

Sal solves a word problem about the weights of TVs and DVDs by creating a system of equations and solving it.

9:00

Systems of equations with elimination: King's cupcakes

Sal uses simple elimination to figure out how many cupcakes are eaten by children and adults.

2:38

Systems of equations with elimination: Sum/difference of numbers

Sal finds two numbers that sum to 70 and have a difference of 24.

9:21

Systems of equations with elimination: potato chips

Sal solves another system of equations using elimination.

4:15

Systems of equations with elimination: coffee and croissants

Sal solves a word problem about the price of coffee and croissant by creating a system of equations and solving it.

5:58

Systems of equations with substitution: coins

Sal solves a word problem about the number of nickles and quarters in a piggy bank by creating a system of equations and solving it.

5:21

Systems of equations with substitution: potato chips

Sal solves a systems of equations with substitution.

8:02

Systems of equations with substitution: shelves

Sal solves a system of equations to figure out how long a couple of different shelves will be.

Exercise

Systems of equations word problems

Solve word problems by modeling them into a system of equations and solving it.

3:43

Age word problem: Imran

Sal solves the following age word problem: In 40 years, Imran will be 11 times as old as he is right now. How old is he right now?

6:28

Age word problem: Ben & William

Sal solves the following age word problem: William is 4 times as old as Ben. 12 years ago, William was 7 times as old as Ben. How old is Ben now?

5:21

Age word problem: Arman & Diya

Sal solves the following age word problem: Arman is 18. Diya is 2. How many years will it take for Arman to be 3 times as old as Diya?

Exercise

Age word problems

Solve age word problems with a system of equations.

## Thinking about solutions to systems

You know how to solve systems of equations (for the most part). This tutorial will take things a bit deeper by exploring cases when you might have no solutions or an infinite number of them.
Common Core Standards: 8.EE.C.8, 8.EE.C.8b

5:29

Systems of equations number of solutions: x+2y=13 & 3x-y=-11

Sal determines whether a given system is consistent or not. Don't know what "consistent system" means? Watch this!

9:25

Systems of equations number of solutions: fruit prices (1 of 2)

Sal gives an example of a system of equations that has no solution!

7:08

Systems of equations number of solutions: fruit prices (2 of 2)

Sal gives an example of a system of equations that has infinite solutions!

5:08

Systems of equations number of solutions: y=3x+1 & 2y+4=6x

Sal graphs a system of equations to find the number of solutions it has.

5:10

Systems of equations number of solutions: 4x+2y=16 & y=-2x+8

Sal determines whether a given system is dependent or not. Don't know what "dependent system" means? Watch this!

6:36

Systems of equations number of solutions: 10x-2y=4 & 10x-2y=16

Sal determines how many solutions the following system of equations has by considering its graph: 10x-2y=4 and 10x-2y=16.

Exercise

Number of solutions to a system of equations graphically

Determine the number of solutions of a given system of equations by considering its graph.

6:21

Forming systems of equations with different numbers of solutions

Sal write an equation that along with the equation 4x + 5y = 2 forms a system of equations with infinitely many solutions.

Exercise

Number of solutions to a system of equations algebraically

Determine the number of solutions of a given system of equations by considering its algebraic solution process.