Systems of Equations
Name ____________________________
2
Writing in Slope-Intercept Form
Learning Target: To write an equation in slope-intercept form in order to graph
Slope-Intercept Form:
y = mx + b
Steps:
 Move y term to the left (if needed) by adding or subtracting
 Move x term to the right (if needed) by adding or subtracting
 Move b term to the right (if needed) by adding or subtracting
 Divide every term by the coefficient (number) in front of y (if needed)
Rewrite each equation in slope-intercept form. Then identify the slope and y-intercept.
1. 3x + y = 9
2. −8x + y = 3
3. 6x + 2y = −10
4. −2x – 3y = 9
Rewrite each equation in slope-intercept form. Then graph.
5.
3x + 2y = 6
6. −2x + 3y = −3
3
Graph the following equations.
7. 7x + y = 4
9.
3y – 6 = 3x
8.
10.
4x – y = 2
9x + 6y = −6
4
Solving Systems by Graphing Notes (day 1)
___________________________________ - Two or more equations in two or more variables.
____________________________________ - Any ordered pair that is a solution to all the equations
in the system.
A system of equations can produce three different results:
Intersecting Lines


Exactly _____ solution
Equations will have different
____________
Parallel Lines


_______ solutions
Equations will have same
___________ but different
_____________
Coinciding Lines


_____________ solutions
Equations will have same
_______________ and
same _________________
5
Determine the number of solutions to the system by examining their equations.
one solution, no solution, infinite solutions.
1. y = 3x – 2
y = 3x + 2
2. y = 4x – 1
y = 2x + 8
3. y = -1x + 6
5. y = -3x – 2
y = 3x - 2
6. y = x + 3
Determine the number of solutions by examining their graphs.
one solution, no solution, infinite solutions.
7.
8.
9.
10.
4. y 
2
x4
3
y
2
x4
3
y = -1x + 6
y=x–2
A solution to a system must be ___________ in all equations.
11.) Is (1, 2) a solution to the
system?
5x + y = 7
x - 3y = 11
12.) Is (2, -3) a solution to the
system?
4x + y = 8
x - 4y = 12
13.) Is (1, 3) a solution to the
system?
y = 2x + 1
x + 2y = 7
6
Graph each system and to find solution.
14. y = x - 3
y = 3x – 7
15.
7
SYSTEMS OF LINEAR EQUATIONS & INTERSECTIONS
Procedures for the TI-83 Graphing Calculator
Recall that the solution is the value (x, y) that will work in both equations. Graphing calculators can
find solutions for linear systems two different ways.
Example: y = 3x – 7 and y = −1/2x + 7 (To enter −1/2 type – 1 ÷ 2)
Using a Table:
1. Go to y = and enter both of your equations; one for Y1 and one for Y2.
2. Go to 2nd window (tblset).
Be sure TblStart = 0, ∆1 = 1, Indepnt: Auto and Depend: Auto
(By selecting Auto the calculator will automatically start at 0 and change by 1. If you
wanted to set the values for x or y you would change this to Ask.)
3. Go to 2nd graph (table).
This will give you a table of values with the x column being the independent value.
Column Y1 gives you the dependent values for the first equation. Column Y2 gives you
the dependent values for the second equation.
4. Scroll up or down until you find the Y1 and Y2 values the same.
5. According to the table, the answer to this equation is ( _____, _____ ).
Using a Graph:
1. Start by setting the screen size (window). Press Zoom then 6:ZStandard.
2. Go to y = and enter the first equation in Y1 and the second equation in Y2 (this should already
be done).
3. Press graph to see both equations graphed.
4. To find the intersection, go to 2nd trace (calc), and select 5:intersect.
Select the first curve by making sure the cursor is on one line- enter. Select the second
curve by making sure the cursor is on the second line- enter. Make a guess by scrolling
left or right close to the point of intersection.
5. According to your graph, the answer to this equation is ( _____, _____ ).
Try these examples using both the table and the graph. Be sure your equations are in slope-intercept
form first.
1. y = 5x − 3
2. y = 2x – 13
3.
x + 4y = 1
y = 3x + 1
y=x–9
x – y = −4
8
Solving Systems by Graphing Notes (day 2)
Graphing a line is easiest when it is in ___________________________________________.
Write each equation in slope-intercept form.
1.
3x + y = 7
2.
-8x + y = 4
3.
4x - y = 2
4.
6.
2x - 5y = 3
2x + 4y = 4
5.
x + 3y = -6
Graph the system of equations and name the solution.
7.
x-y=3
x - 2y = 3
8.
3x - y = 7
3x - 2y = 8
9
9.
-x + y = 2
11.
2x + 3y = 9
x - y = -2
x = -3
How do you graph a line with a y-variable only?
Ex. y = 2
10.
y = 2x + 3
3y - 6x = -6
10
Substitution Day 1
Steps:
1. Solve for one of the variables. Hint: Choose a variable with a coefficient of 1.
2. Substitute expressions into other equation and solve.
3. Substitute answer into first equation and solve.
4. Write answer as an ordered pair (x, y).
1. y = 6
2x + 3y = −20
2. x = 4y
4x – y = 75
3. −2x + 4y = 6
y = 2x – 3
4. 5x – y = −9
x = 4y + 2
5. y = x – 7
2x – 4y = 4
6. y = 2x – 5
3x + 2y = −3
11
7. 3x – 4 = y
2x – 3y = −9
3 possible outcomes when you graph a system on a coordinate plane:
Intersecting Lines
Parallel Lines
Same Line
8. y = −3x + 13
6x + 2y = 26
9.
5x + y = 3
y = −5x + 6
12
Substitution Day 2
Steps:
1. Solve for one of the variables. Hint: Choose a variable with a coefficient of 1.
2. Substitute expressions into other equation and solve.
3. Substitute answer into first equation and solve.
4. Write answer as an ordered pair (x, y).
1. 4x + y = 12
−2x – 3y = 14
2. 2x + 2y = 8
x + y = −2
3. 2x + 2y = 3
4y – x = 1
4. 3x – y = 4
2x – 3y = −9
5. 2y – x = 2
x – 2y = 8
6. −4x = 2y + 24
2x + y = −2
13
Elimination Day 1
1.
3.
5.
7.
x – 3y = 7
3x + 3y = 9
2x – 3y = 14
x + 3y = −11
3x – y = −9
−3x – 2y = 0
2a + 2b = −2
3a – 2b = 12
2. x + y = −4
x– y=2
4.
−3x – 4y = −1
3x – y = −4
6.
3x + y = 4
2x – y = 6
8.
−0.2x + y = 0.5
0.2 x + 2y = 1.6
14
A little twist……. Multiply by −1
9. 2x – 3y = 11
5x – 3y = 14
10.
6x + 5y = 4
6x – 7y = −20
11. 3x – 4y = −14
3x + 2y = −2
12.
3x + y = 1
x+y=3
13.
−3x – 4y = −23
−3x + y = 2
14.
x – 2y = 6
x+y=3
15.
5m – n = 6
5m + 2n = 3
16.
4x + 2y = 6
4x + 4y = 10
17. The sum of two numbers is 29 and their difference is 15. What are the numbers?
15
Elimination Day 2
3x – y = 2
x + 2y = 3
1.
2x + 3y = 4
−x + 2y = 5
2.
3.
4x – y = 9
5x + 2y = 8
4. 2x + 3y = 6
x + 2y = 5
5.
3x – 4y = −4
x + 3y = −10
6.
4x + 5y = 6
6x – 7y = −20
7.
2x – 3y = 42
3x + 2y = 24
8.
4x – 3y = 22
2x – y = 10
16
9.
11.
6x + 2y = 20
−2x + 4y = −16
6x – 4y = −8
4x + 2y = −3
10.
3x – 2y = −7
2x – 5y = 10
12. Eight times a number plus five times another
number is −13. The sum of two numbers is 1.
What are the numbers?
17
Systems 3 Ways and Applications
You have learned 3 ways to solve a system of equations…
Graphing
Substitution
All three methods will give you the ______________ answer!!!!
1. Solve using all three methods:
y = -x + 3 and x – y = -1
2. Determine the best method to solve
the system of equations. Then solve the
system.
x + 5y = 4
3x - 7y = -10
Elimination
18
Solve each system using the method you selected on previous page.
3.
x=y+8
4x + 2y = 2
4. y = -x + 14
-x + y = 4
5.
3x + 2y = 11
4x + 5y = 3
6.
m=p+7
3m - 5p = 25
7.
3x - 4y = 1
4x - 5y = 3
8.
4x - 2y = 38
x + 2y = 7
9. The sum of two numbers is 48, and their
difference is 24. What are the numbers?
10. The difference between the length and width
of a rectangle is 7 cm. Find the dimensions of the
rectangle if its perimeter is 50 cm.
19
Problem Solving Day 1



You can solve many real world problems using a system of equations.
To do this, you need to write a system of equations using the given information.
Look for two different details of information given. Often, these are given in two different
sentences.
Ex. Chelsea and Zack are both dog sitters. Chelsea charges $2 per day plus a sign-up fee of $3.
Zack charges a flat rate of $3 per day. Write a system of linear functions (equations) to represent the
amount each of them charge, y for any number of days, x.
Let: ________ =
________ =
Chelsea’s Fee
Zack’s Fee
Graph each equation to determine after how many days Chelsea and Zack earn the same amount for
dog sitting.
20
Identify and interpret the solution.
Verify that the intersection of the two equations is the
solution to the system.
If you were going on vacation and needed
to hire a dog sitter. Which person would
you hire, Chelsea or Zack? Why?
Solve the system of equations by either substitution or
elimination. What is the solution? Compare to your
answer to the intersection of the graph.
21
Problem Solving Day 2
1. The sum of two numbers is 36. The difference 2. The rectangle has a perimeter of 18 cm. Its
of the same two numbers is 6. Find the numbers. length is 5 cm greater than its width. Find the
dimensions.
3. The volleyball club has 41 members. There
are 3 more boys than girls. How many boys and
how many girls are there?
4. A theater sold 900 tickets to a play. Floor
seats cost $12 each and balcony seats are $10
each. Total receipts from the tickets were $9780.
How many of each type of ticket were sold?
22
5. At the local deli, the cost of 3 pizzas and 4
sandwiches is $68. The cost of 3 pizzas and 7
sandwiches is $92. What is the cost of a pizza
and a sandwich?
6. Ben has nickels and dimes in his toy bank.
He has a total of 45 coins and the total value of
the coins is $3.60. How many coins of each kind
does he have?
Ex. 7 Joey “Goodfella” Branzolli likes to shuffle music as a disk jockey. He had 80 CDs, which ere
rock and rap CDs. He bought his CDs as a bulk rate: $4 for rock and $3 for rap CDs. If his collection
was worth $292, how many of each type of CD did he own?