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Systems of Equations
OBJECTIVES• To understand what a system of
equations is. • Be able to solve a system of
equations from graphing the equations
• Determine whether the system has one solution, no solution, or an infinite amount of solutions.
• Be able to graph equations without using a graphing calculator.
Defining a System of Equations
• A grouping of 2 or more equations, containing one or more variables.
x + y = 2x + y = 2
2x + y = 52x + y = 5
2y = x + 22y = x + 2
y = 5x - 7y = 5x - 7
6x - y = 56x - y = 5
How do we “solve” a system of
equations??? • By finding the point where two or
more equations, intersect.
x + y = 6x + y = 6
y = 2xy = 2x Point of intersectionPoint of intersection
66
44
22
11 22
How do we “solve” a system of
equations??? • By finding the point where two or
more equations, intersect.
x + y = 6x + y = 6
y = 2xy = 2x (2,4)(2,4)
66
44
22
22
11
ax + by = c
2x + 3y = 62x + 3y = 6 ax + by = cax + by = c
-2x-2x -2x-2x 3y = 6 - 2x3y = 6 - 2x
33 33 33
y = 2 -y = 2 - 2233
xx
y = - + 2y = - + 22233
xx y = mx + by = mx + b
WE WANT THIS FORM!!!WE WANT THIS FORM!!!
(Standard Form)(Standard Form)
(Slope- Intercept)(Slope- Intercept)
Non-Unique Solutions
No SolutionNo Solution: : • when lines of a graph are parallelwhen lines of a graph are parallel
• the slopes are the same and the the slopes are the same and the equation must be in slope-intercept formequation must be in slope-intercept form
• since they do not intersect, there is no solutionsince they do not intersect, there is no solution
Example of No Solution
2x+y=82x+y=8y=-2x-3y=-2x-3
Remember when Remember when graphing the graphing the equations need to be equations need to be in slope intercept in slope intercept form!!!form!!!
Infinite SolutionsInfinite Solutions: :
Non-Unique Solutions
• a pair of equations that have the a pair of equations that have the same slope and y-intercept.same slope and y-intercept.
• also call a also call a Dependent SystemDependent System
Example of Infinite Solutions
2x+4y=82x+4y=84x+8y=164x+8y=16
Unique Solutions
One SolutionOne Solution: : • the lines of two equations intersectthe lines of two equations intersect
• also called an also called an Independent SystemIndependent System
Graphing Systems in Slope-Intercept
Form• 2y + x = 82y + x = 8
y = 2x + 4y = 2x + 4Steps:Steps:
1. Get both equations in slope-intercept form1. Get both equations in slope-intercept form
2. Find the slope and y intercept of eq. 1 then graph2. Find the slope and y intercept of eq. 1 then graph
3. Find the slope and y intercept of eq. 2 then graph3. Find the slope and y intercept of eq. 2 then graph
4. Find point of intersection4. Find point of intersection
(0,4)(0,4)
€
y =−1
2x + 4
y = 2x + 4y = 2x + 42y + x = 82y + x = 8
-x -x-x -x
2y= 8 -1x2y= 8 -1x
2 2 22 2 2
Equation 1Equation 1
m= -1/2m= -1/2b=4b=4
Equation 2Equation 2
m= 2/1m= 2/1B=4B=4
The solution The solution to the to the system is system is (0,4)(0,4)
Example 1: 2y+x=8 and y=2x+4Example 1: 2y+x=8 and y=2x+4
Notice the slopes and Notice the slopes and y intercepts are y intercepts are different so there will different so there will only be one solutiononly be one solution
Example 2: y=-6x+8 and y+6x=8Example 2: y=-6x+8 and y+6x=8Equation 1Equation 1
y=-6x+8y=-6x+8
b=8b=8
m=m=-6(down)-6(down) 1(right)1(right)
Equation 2Equation 2
y+6x=8y+6x=8
-6x -6x-6x -6x
y=8-6xy=8-6x
m=m=-6(down)-6(down) 1(right)1(right)
b=8b=8Notice both Notice both equations have the equations have the same intercept and same intercept and slope. This means slope. This means all the points are all the points are solutions.solutions.
Infinite SolutionInfinite Solution
Example 3: x-5y=10 and -5y=-x+40 Example 3: x-5y=10 and -5y=-x+40
Equation 1Equation 1
x-5y=10x-5y=10--x -xx -x
--5y=10-1x5y=10-1x__ __ __ __ __ __
-5 -5 -5-5 -5 -5
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y = −2 +1
5x
b=-2b=-2
m=1/5m=1/5
Equation 2Equation 2
-5y=-x+40-5y=-x+40__ __ _____ __ ___
-5 -5 -5-5 -5 -5
€
y =1
5x −8
b=-8b=-8
m=1/5m=1/5
The slopes are the The slopes are the same with different same with different intercepts so the intercepts so the lines are parallellines are parallel No SolutionNo Solution
You Try ExamplesDetermine whether the following equations have one, none, or Determine whether the following equations have one, none, or infinite solutions by graphing on the graph paper provided.infinite solutions by graphing on the graph paper provided.
2)2)y = x - 1y = x - 1
y = 3y = 3
2233
3)3) x + 2y = 6x + 2y = 6
x + 2y = 8x + 2y = 8
1)1)
ANS:ANS: One Solution One Solution (6,3)(6,3)
ANS:ANS: No Solution No Solution
ANS:ANS: Infinite Solutions Infinite Solutions€
1
3x +1
8y =1
€
8x + 3y = 24
€
1
3x +1
8y =1
Slope Intercept FormSlope Intercept Form
€
y =−8
3x + 8
Equation 1Equation 1
Equation 2Equation 28x+3y=248x+3y=24
€
y =−8
3x + 8
Infinite Infinite SolutionSolution
Equation1Equation1
€
x + 2y = 6
y =−1
2x + 3
Equation 2Equation 2
€
x + 2y = 8
y =−1
2x + 4
Notice the slopes are the same Notice the slopes are the same and the y-intercepts are and the y-intercepts are different. This means that the different. This means that the lines are parallel so they will lines are parallel so they will never intersectnever intersectNo SolutionNo Solution
Equation 1Equation 1
€
y =2
3x −1
Equation 2Equation 2
y=3y=3
Notice the Notice the equations have equations have different slopes different slopes and y-intercepts and y-intercepts so the lines will so the lines will have 1 solutionhave 1 solutionSolution (6,3)Solution (6,3)
Algebraically determine if a point is a solution
We must always verify a proposed solution algebraically. We propose (1,6) as a solution, so now we plug it in to both equations to see if it works:
y = 6x and y = 2x + 4, 6 = 6(1) and 6=2(1)+4, (6)= 6 and 6= 6.
Yes, (1,6) Satisfies both equations!
Algebraically determine if a point
is a solution• Determine if (2,12) is a solution to
the system.• y=6x and y=2x+4• 12=6(2) and 12=2(2)+4• 12=12 and 12=8• Yes No• (2,12) is NOT a solution to the system
