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Published byMae Miles Modified over 9 years ago
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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:
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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
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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.
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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 = 7substitute (x+3) for y 3x – (x+3) = 7distribute -1 through the parentheses 3x – x – 3 = 7simplify 2x – 3 = 7simplify 2x = 10simplify x=5
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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:
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If x = 5 y = x + 3 y = 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.
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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 = 5y = 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 = 20 3x + 9y = 45
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Try this one… 4x – 12y = 20 3x + 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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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 – 3yplug in (5/3) for y x = 15 – 3(5/3)simplify x = 15 – 53 times (5/3) is 5 x = 10
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x = 10y = 5/3 So our solution is (10, 5/3) Always check your solution!
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To review, 4x – 12y = 20 3x + 9y = 45 We simplified one of the equations 3x + 9y = 45 x = 15 – 3y Plugged this “15 – 3y” in for x in the other equation 4x – 12y = 20 4(15 – 3y) – 12y = 20 Solved for y y = 5/8 Plugged in this y-value into the other equation to find x x = 15 – 3(5/3) x = 10
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