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Algebra 2
Systems of Equations
Systems of Equations
• What is a system of equations?
• What does it mean to solve a system of
equations?
Ways to Solve Systems of Equations
• Graphing
• Substitution
• Elimination
In order to solve a system of equations
graphically you typically begin by
making sure both equations are in
slope-intercept form.
y  mx  b
Where m is the slope and b is the y-intercept.
Examples:
y = 3x- 4
Slope is 3 and y-intercept is - 4.
y = -2x +6
Slope is -2 and y-intercept is 6.
What information do you need to graph the
following equations?
•y = 2x +1
• 3x –y = -2
• x + 2y = 4
y  2x 1
x y 5
x  y  1
2y  x  4
 2x  3 y  3
x  6 y  24
1
y   x 1
2
x  2y  6
 x  2y  3
4 y  2x  8
1
y  x2
3
2 x  6 y  12
1
y   x3
4
x  4 y  12
Lesson 3.2
Solving Systems of Equations
Algebraically
Solving Systems of Linear Equations by Substitution
1. Solve for a variable
2. Substitute for that variable in the other
equation
3. Solve this equation for the remaining variable
4. Put your solution back into either of the
original equations to solve for the other
variable
5. Check your solution with the other equation
Solving Systems of Linear Equations by Substitution
SPECIAL CASES
• If when using substitution both variables drop out
and you get something like: 10=6
This statement is not true, so there is no solution
(the lines are parallel lines)
• If when using substitution both variables drop out
and you get something like: 10=10
This statement is true so there are infinite
solutions (the lines are the same lines)
Solving Systems of Linear Equations by Substitution
y  2x
x y 6
y  x 1
x  3 y  12
2y  x  5
2x  3y  7
Chatty Phone charges a flat monthly fee of $20 plus 8¢ a minute.
Telco charges $14 plus 10¢ a minute. When do they charge the
same?
Systems of Equations
WORD PROBLEMS
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
Two small pitchers and one large pitcher can hold 8 cups of
water. One large pitcher minus one small pitcher constitutes 2
cups of water. How many cups of water can each pitcher hold?
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
A test has twenty questions worth 100 points. The test consists of
True/False questions worth 3 points each and multiple choice
questions worth 11 points each. How many multiple choice
questions are on the test?
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
Bill and Steve decide to spend the afternoon at an amusement park
enjoying their favorite activities, the water slide and the gigantic
Ferris wheel. Their tickets are stamped each time they slide or
ride. At the end of the afternoon they have the following tickets:
Fun Time Amusements
Fun Time Amusements
Water Slide:
Water Slide:
Ferris Wheel:
Ferris Wheel:
Total: $17.70
Total: $15.55
Bill's Ticket
Steve's Ticket
How much does it cost to ride the Ferris Wheel?
How much does it cost to slide on the Water Slide?
Kristin spent $131 on shirts. Fancy shirts cost $28 and plain shirts
cost $15. If she bought a total of 7 then how many of each kind did
she buy?
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
There are 13 animals in the barn. Some are chickens and some are
pigs. There are 40 legs in all. How many of each animal are there?
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
At Elisa's Printing Company LLC there are two kinds of printing
presses: Model A which can print 70 books per day and Model B
which can print 55 books per day. The company owns 14 total
printing presses and this allows them to print 905 books per day.
How many of each type of press do they have?
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
Four times one number added to another number is 36. Three
times the first number minus the other number is 20. Find the
numbers.
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
One number added to three times another number is 24. Five times
the first number added to three times the other number is 36. Find
the numbers.
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
Eight times a number plus five times another number is 13. The
sum of the two numbers is 1. What are the numbers?
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
A library contains 2000 books. There are 3 times as many nonfiction books as fiction books. Write and solve a system of
equations to determine the number of nonfiction and fiction
books.
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
How do the quarter Pounder and Whopper with cheese measure up
in the calorie department? Actually, not too well. Two Quarter
Pounders and three Whoppers with cheese provide 2607 calories.
Even one of each provide enough calories to bring tears to Jenny
Craig’s eyes-9 calories in excess of what is allowed on a 1000
calories-a day diet. Find the calories in each item.
•Define your variables.
•Write your system of equations.
•Solve the system of equations.
Solving Systems of Linear Equations by Elimination
1. Write equations in standard form (variables line up)
2. Multiply one of the equations to get coefficients of
one of the variables to be opposites
3. Add (or subtract) equations – so that one variable
drops out
4. Solve for the remaining variable.
5. Plug you solution back into one of the original
equations and solve for the other variable.
2x  y  6
3x  y  4
x  3y  5
x y 3
y  3x  4
2x  y  5
2 x  3 y  4
x  4y  9
 2x  3 y  5
5x  2 y  4
Systems of Equations
Applications
3.2 Applications of Linear
Systems of Equations
1. Determine what you are to find – assign variables
2. Draw a diagram, figure or make a chart of
information.
3. Write the system of equations
4. Solve the system using substitution or elimination
5. Answer the question from the problem.