solving systems of linear equations wait a minute! what’s a system of linear equations? –a...
TRANSCRIPT
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Solving Systems of Linear Equations
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Wait a minute!
• What’s a system of linear equations?– A system is a set of linear sentences which
together describe a single situation.
• How do I know if I have a solution?– A solution to a system is a pair of numbers
that satisfies all equations within the solution.
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Examples of Linear Systems
x y 22x y 8
y 9xy 2x 7
The system at the upper left shows an example of two linear equations in standard form (Ax+By=C).
The system at the lower left shows an example of two linear equations in slope-intercept form (y=mx+b).
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Non-examples of Linear Systems
y 1
2x 2
y 12x 1
x y 3
The situation in the upper left is a system of equations, but it is not a system of linear equations because the top equation is a quadratic.
The situation at the lower left is not a system because it is not a set of linear sentences. It is only one linear sentence.
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Which of the following is a system of linear equations?
y x2x 3y 15
y 2x 3
y 1x
3
*Note: The equations are brought in as objects from Equation Editor. Therefore, press the button in the blue space below the equations.
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Be Careful…
• This is a system of equations, but both equations are not linear.
• Linear equations always have x raised to the first power.
1
xx 1
Review Examples of Linear Systems
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Great Job!
Now that you can identify a system of linear equations, it is time to learn how to solve them!
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Solving Systems of Linear Equations
Solving by Graphing
Solving by Substitution
Solving by Elimination
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Steps to Solve by Graphing
1. Graph both equations in the system.2. Find the ordered pair at the point of
intersection.3. This ordered pair is the solution to the
system! (It contains the x and y values that will make both equations true!) Always check your solution in the equations.
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Important Note:
It is VERY important to graph your lines accurately! Use graph paper and a straight edge!
Let’s look at an example!
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Solve
63
12
xy
xy
First, graph y=2x+1. Remember, plot the y-intercept first (1) and then use the slope (2) to find another point on the line.
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Solve
63
12
xy
xy
Second, graph y=-3x+6 on the same set of axes.
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The solution is the point of intersection!
The lines intersect at (1,3). This means that when x=1, then y=3 in BOTH equations in the system. To be sure, our next step is to check the solution in the equations.
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Does (1,3) work in both equations? Let’s check!
33
123
1123
12
xy
33
633
6133
63
xy
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(1,3) is the solution to
63
12
xy
xy
Since (1,3) is the point of intersection and it worked in both equations of the system, this is the solution!
Test your understanding.
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Solve
1532 yx
xy
The first step is to graph both of these equations.
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Solve
1532 yx
xy
Choose the graph below that has both equations graphed correctly. (Click in the yellow area below the graph!)
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Now check your solution!
11 xy
151
1532
1513)1(2
1532
yx
The intersection point is (-1,1). Does this work in both equations?
11 xy
The solution does not work in either equation! This means the graph is not correct!
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Now check your solution!
22 xy
1510
1564
15)2(3)2(2
1532
yx
The intersection point is (-2,-2). Does this work in both equations?
The solution does not work in the second equation! This means the graph is not correct!
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Now check your solution!
1515
1596
15)3(3)3(2
1532
yx
33 xy
The intersection point is (-3,-3). Does this work in both equations?
Great job! (-3,-3) works in both equations so it is the solution!
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Solving by Substitution
Look at the system
278
255
xy
xy
278255 xyx
We can say that
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Solving by Graphing
• You have done a great job solving systems of equations by graphing.
• There are two other techniques to solving systems of linear equations. – Solving by Substitution– Solving by Elimination
• Return to the Home Menu to learn more!
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By the Transitive Property…
• The Transitive Property states that if a=b and b=c, then a=c.
• From our system, we know that 5x-25=y and y=-8x+27.
• Therefore, 5x-25=-8x+27!– This is an equation with one variable. We
can solve for x!
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Solve the Equation
278255 xxx8 x8
272513 x25 25
5213 x13 13
4x
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X=4 is part of the solution!
• When x=4, both equations will result in the same y value. This is the other coordinate in our solution!
• Substitute x=4 into one of the equations. (It does not matter which equation you use; both will give the same result for y.)
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Let’s find the y value!
5
2520
2545
255
y
y
y
xy
5
2732
2748
278
y
y
y
xy
Regardless of which equation you use, when x=4 then y=-5. Therefore, the solution to this system is (4,-5).
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Rules for Solving by Substitution
1. Solve both equations for y.
2. Set the equations equal to each other.
3. Solve for x.
4. Substitute the x value into one of the equations to solve for y.
5. Once you have x and y, write your solution as an ordered pair.
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Try this example.
Using substitution, solve
104
3
52
1
xy
xy
Set the equations equal to each other and solve for x. When you do this, what result do you get?
X=3 X=5 X=12
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If x=3, solve for y.
5.32
72
10
2
3
52
3
532
1
52
1
y
y
y
y
y
xy
Does the solution (3,-3.5) check in both equations?
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Check the solution (3,-3.5)
5.35.3
55.15.3
532
15.3
52
1
xy
75.75.34
315.3
4
40
4
95.3
104
95.3
1034
35.3
104
3
xy(3,-3.5) does not work in the second equation. Therefore, this is not the solution.
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If x=5, solve for y.
5.2
55.2
552
1
52
1
y
y
y
xy
Does the solution (5,-2.5) check in both equations?
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Check the solution (5,-2.5)
5.25.2
55.25.2
552
15.2
52
1
xy
25.65.24
255.2
4
40
4
155.2
104
155.2
1054
35.2
104
3
xy(5,-2.5) does not work in the second equation. Therefore, this is not the solution.
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If x=12, solve for y.
1
56
5122
1
52
1
y
y
y
xyDoes the solution (12,1) check in both equations?
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Check the solution (12,1)
11
561
5122
11
52
1
xy
11
1091
10124
31
104
3
xyGreat job! Since (12,1) works in both equations, it is the solution to this system!
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Solving by Substitution
• You have done a great job solving systems of equations by substitution.
• There are two other techniques to solving systems of linear equations. – Solving by Graphing– Solving by Elimination
• Return to the Home Menu to learn more!
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Solving by Elimination
• We know that if a=b and c=d, then a+c=b+d.– Apply this to equations to solve by
elimination.
• The goal is to add the equations of a system to get a variable to cancel out.
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Solve
150
120
yx
yx
150
120
yx
yx
2702 x
First, add this equations together. Since there is –y in the first equation and +y in the second equation the y variable will cancel out!
2 2135x
Now that we know what x is, we can solve for y!
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If x=135, solve for y!
150135
150
y
yx
13515y
You can use either equation to solve for y.
135
Our solution is (135,15). To verify that this is the solution, we can check the coordinates in both equations.
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Check the solution (135,15)
120120
12015135
120
yx
150150
15015135
150
yx
Since (135,15) works in both equations, it is the solution to this linear system! Now try an elimination problem on your own…
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Solve
642
282
yx
yx
When you add the equations in the system together, which variable will you end up solving for?
x y
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Check again…
• When you add the equations, 2x+-2x will cancel out, leaving you y to solve for.
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Good job!
Now solve for y! What is the result?
Y=2/3 Y=1 Y=2
150
120
yx
yx
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If y=2/3, solve for x.
3
53
102
3
6
3
162
23
162
23
282
282
x
x
x
x
x
yx
Does the solution check in both equations?
3
2,3
5
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Check the solution
3
2,3
5
22
23
6
23
16
3
10
23
28
3
52
282
yx
63
2
63
8
3
10
63
24
3
52
642
yx
This ordered pair does not work in the second equation. Therefore, it is not the solution to the system.
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If y=1, solve for x.
3
62
282
2182
282
x
x
x
x
yxDoes the solution (-3,1) check in both equations?
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Check the solution (-3,1)
22
286
21832
282
yx
62
646
614)3(2
642
yx
This ordered pair does not work in the second equation. Therefore, it is not the solution to the system.
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If y=2, solve for x.
7
142
2162
2282
282
x
x
x
x
yxDoes the solution (-7,2) check in both equations?
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Check the solution (-7,2)
22
21614
22872
282
yx
66
6814
624)7(2
642
yx
Good job! This ordered pair works in both equations. Therefore, (-7,2) is the solution to the system!
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Solving by Elimination
• You have done a great job solving systems of equations by elimination.
• There are two other techniques to solving systems of linear equations. – Solving by Graphing– Solving by Substitution
• Return to the Home Menu to learn more!• If you have already learned all three methods,
click on the blue arrow for more information!
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Solving Systems of Equations
You have now learned the basic principles to solving systems of linear equations using the three methods: graphing, substitution, and elimination.
For more practice on solving systems of linear equations (or to look at more advanced examples) click here.
Back to the beginning… End