SOLVING LINEAR SYSTEMS WITH SUBSTITUTION by Sam Callahan.
Transcript of SOLVING LINEAR SYSTEMS WITH SUBSTITUTION by Sam Callahan.
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SOLVING LINEAR SYSTEMS WITH SUBSTITUTION
by Sam Callahan
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By now you’ve learned to solve systems of equations using graphing and finding where the lines intersect:
73 yx
3xy
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The problem with solving by graphing though, is evident when you look at graphs like the one below.
This solution (the blue point
where the lines intersect) isn’t
on a gridline and very
hard to accurately identify
3xy 73 yx
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Although graphing is simple and visual, it is really only accurate enough to use with systems that have integer answers.
Substitution is a way to solve systems of equations analytically, or without graphing.
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If we take the same system of equations:
We can make solving these equations possible by working with one variable at a time. To do this, we substitute one side of the equation in for the other variable.
3xy
73 yx
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If we know that y = x+3, we can plug in (x+3) in for y in the second equation
to find out what x is.
3x – y = 7 substitute (x+3) for y
3x – (x+3) = 7 distribute -1 through the parentheses
3x – x – 3 = 7 simplify
2x – 3 = 7 simplify
2x = 10 simplify
x=5
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x = 5
Now that we know what x is (5), we can plug 5 in for x in either equation to find out what y is. I like to use whichever equation has simpler numbers to work with.
In this case, that equation is: 3xy
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If x = 5
y = x + 3y = 5 + 3
y = 8
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x = 5 and y = 8, so our solution to the system
In (x, y) form is (5, 8)
Now let’s check our answer.
3xy
73 yx
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Checking your solution
You should check your answer using the equation that you didn’t just solve.
For example, my last step was plugging in 5 for x into y = x + 3
I should check with the other equation, 3x – y = 7
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Checking your solution
x = 5 y = 8
3x – y = 7 plug in your values for x and y
3(5) – (8) = 7 simplify
15 – 8 = 7 simplify
7 = 7 make sure your statement is true
We ended up with a true statement, so our solution works!
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Try this one…
4x – 12y = 203x + 9y = 45
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Try this one…
4x – 12y = 203x + 9y = 45
Unlike the previous example, we aren’t given an equation right away that says what x or y
is equal to, so we have to simplify one of these equations so that it reads
y=_____ or x=______
Choose one of the equations to simplify.I’ll use 3x + 9y = 45
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3x + 9y = 45
You can start with either variable, but I want to solve for x first because I don’t want a fraction that would result if I divided everything by 9.
3x + 9y = 45 subtract 9y to put it on the right side of the equation
3x = 45 – 9y divide everything by 3 so that x will be on its own
/3 /3 /3
x = 15 – 3y
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x = 15 – 3y
Now we have what x is equal to, so we can plug in“15 – 3y” for x in the other equation.
4x – 12y = 20 plug in the expression for x
4(15 – 3y) – 12y = 20 distribute
60 – 12y – 12y = 20 simplify
60 – 24y = 20 simplify
40 = 24y simplify
y = 5/3 I used a calculator for this step, but you could also simplify this fraction (40/24) by hand
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y = 5/3
Plug this y-value into the other equation we found (x = 15 – 3y) to find x.
x = 15 – 3y plug in (5/3) for yx = 15 – 3(5/3) simplifyx = 15 – 5 3 times (5/3) is 5
x = 10
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x = 10 y = 5/3
So our solution is (10, 5/3)
Always check your solution!
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To review, 4x – 12y = 203x + 9y = 45
We simplified one of the equations3x + 9y = 45 x = 15 – 3y
Plugged this “15 – 3y” in for x in the other equation4x – 12y = 204(15 – 3y) – 12y = 20
Solved for yy = 5/8
Plugged in this y-value into the other equation to find xx = 15 – 3(5/3)x = 10