Download Chapter 5 Notes

Name _____________________________
Algebra I
Date ___________
Period ___________
Systems of Equations
Let’s Review: How do we graph an equation?
1) Place the equation in y-intercept form.
2) Graph the y-intercept.
3) Use the slope to get a second point.
4) Connect the points.
Graph: -2x+3y= -18
Defining a System of Equations
 A grouping of ______________________________, containing one or more
variables.
x+y=2
2x + y = 5
2y = x + 2
y = 5x - 7
6x - y = 5
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Name _____________________________
Date ___________
Algebra I
Period ___________
How do we “solve” a system of equations???
 By finding the point where two or more equations ________________.
x+y=6
y = 2x
6
We also need to verify that our
solution satisfies both equations.
Point of intersection
4
2
1 2
Determining whether a solution works…
1) Plug the point into both equations
2) Solve each equation to make sure both sides are =
Example:
#1) Is (2, -3) a possible answer for 4x+8y=-16 and 5x+ 3y=2?
#2) Is (-1, 3) a solution for 5x+4y=7 and 2x-8y=-26?
Graphing to Solve a Linear System
Let's summarize! There are 4 steps to solving a linear system using a graph.
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Name _____________________________
Algebra I
Step 1: Put both equations in y intercept form.
Date ___________
Period ___________
Solve both equations for y, so that
each equation looks like
y = mx + b.
Step 2: Graph both equations on the
same coordinate plane.
Use the slope and y - intercept for
each equation in step 1. Be sure to
use a ruler and graph paper!
Step 3: Estimate where the graphs
intersect.
This is the solution! LABEL the
solution!
Step 4: Check to make sure your
solution makes both equations true.
Substitute the x and y values into both
equations to verify the point is a
solution to both equations.
Solve the following system of equations by graphing.
Example #1) 2x + 2y = 3
x – 4y = -1
Example #2) 2x+3y=18
-4x+2y=-4
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Name _____________________________
Algebra I
Example #3) 5x+2y=10
15x+6y=30
Date ___________
Period ___________
Example #4) -2x+y=3
-2x+y=1
Types of Solutions
No Solution:
 when lines of a graph are _______________
 since they do ________________________,
there is no solution
 also called an ________________________
Infinite Solutions:
 a pair of equations that have the same
____________________ and
____________________.
 also call a ___________________________
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Name _____________________________
Algebra I
Unique Solution:
 the lines of two equations
___________________
 also called an ____________________
Date ___________
Period ___________
Determine whether the following have a unique, no solution, or infinite
solutions by looking at the slope and y-intercepts
#1)
#2)
#3)
x2y + x = 8
y = -6x + 8
y = 2x + 4
y + 6x = 8
5y = 10
-5y = -x +6
Solving Equations by Graphing on the Calculator
1) Make sure both equations are in y-intercept form
2) Hit y=
3) Input both equations
4) Hit graph
5) Hit 2nd Calc
6) Choose #5: Intersect
7) Hit Enter until it gives you the intersection
#1) y=7+2.5x
y=35.9-6x
#2) y=25+30x
y= 15+32x
#3) y=4x-5.5
y= -3x+5
#4) 2x+y=9
3x+y=16.3
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Name _____________________________
Algebra I
Date ___________
Period ___________
2 Methods for Solving Algebraically
1. Substitution Method: (used mostly when one of the equations has a variable with
a __________________________________________)
2. Elimination Method
Substitution Method
1. Solve one of the given equations for one of the variables. (whichever is the easiest
to solve for)
2. Substitute the expression from step 1 into the other equation and solve for the
remaining variable.
3. Substitute the value from step 2 into the original equation and solve for the 2nd
variable.
4. Check your solution.
5. Write the solution as an ordered pair (x,y).
Ex #1: Solve using substitution method
3x-y=13
2x+2y= -10
1. Solve the 1st eqn for y.
3x-y=13
-y= -3x+13
y=3x-13
2. Now substitute 3x-13 in for the y in the 2nd equation.
2x+2(3x-13)= -10
Now, solve for x.
2x+6x-26= -10
8x=16
x=2
3. Now substitute the 2 in for x in for the equation from step 1.
y=3(2)-13
y=6-13
y=-7
4. Solution: (2,-7)
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Name _____________________________
Algebra I
Date ___________
Period ___________
5. Plug in to check solution.
#2)
y  2x  2
2x  3y  10
Step 1: Solve one equation for one variable.
Step 2: Substitute the expression from step one into the other equation and solve the
equation.
Step 3: Substitute back into either original equation to find the value of the other
variable.
Step 4: Check solution.
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Name _____________________________
Algebra I
#3) y = 4x
3x + y = -21
Date ___________
Period ___________
#4) x + y = 10
5x – y = 2
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Name _____________________________
Algebra I
2a  3b  7
#5)
Date ___________
Period ___________
2a  b  5

Elimination Method
1. Multiply one or both equations by a real number so that when the equations are
added together one variable will cancel out.
2. Add the 2 equations together. Solve for the remaining variable.
3. Substitute the value form step 2 into one of the original equations and solve for
the other variable.
4. Write the solution as an ordered pair (x,y).
Ex #1): 2x-6y=19
-3x+2y=10
1. Multiply the entire 2nd eqn. by 3 so that the y’s will cancel.
2x-6y=19
-9x+6y=30
2. Now add the 2 equations.
-7x=49
and solve for the variable.
x=-7
3. Substitute the -7 in for x in one of the original equations.
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Name _____________________________
Algebra I
2(-7)-6y=19
-14-6y=19
-6y=33
y= -11/2
4. Now write as an ordered pair.
(-7, -11/2)
Date ___________
Period ___________
5. Plug into both equations to check.
#2)
x+y=7
x-y=3
#3)
2x - y = - 8
2x + 2y = 16
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Name _____________________________
Algebra I
#4)
4x+5t=22
5x-t=13
Date ___________
Period ___________
#5)
4x + 2y = 24
- 3x + 6y = - 63
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Name _____________________________
Algebra I
#6) 9x-3y=15
-3x+y= -5
Date ___________
Period ___________
Both equations are for the same line! ______________________
Means any point on the line is a solution.
#7) 6x-4y=14
-3x+2y=7
It means the 2 lines are parallel._________________________________
Since the lines do not intersect, they have no points in common.
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Name _____________________________
Algebra I
#8) 3x-2y=4
2x+y=4
Date ___________
Period ___________
#9) 3c-8d=7
c+2d=-7
#10) 2c-7d=41
6c+5d=-7
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Name _____________________________
Algebra I
#11) 2x + 4y = 50
x - 8y = - 65
Date ___________
Period ___________
#12) 5x - 5y = - 25
3x - 15y = - 39
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