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Question 85022: Solve the system by addition or substitution.
3x – 4y = 8
6x – 2y = 10
Found 2 solutions by bucky, jim_thompson5910:
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given:
.
+3x – 4y = 8
+6x – 2y = 10
.
Let's solve this set by addition. By addition we are going to be adding the two equations
in vertical columns. And we need to get one of the variables in one equation to cancel the
the same variable in the other equation. What that means is that for the variable in one
equation to equal the same variable in the other equation, but have the opposite sign.
.
A little difficult to explain, but easy to do. For example, let's set the goal of getting
the y terms to cancel out. Notice in the top equation that the y term is -4y. If we had
+4y in the bottom equation, it would cancel the -4y of the top equation when we add the
two equations together. We can make the y term in the bottom equation equal +4y if we multiply
it by -2. But if we do that, we also have to multiply all the terms in the bottom equation
by -2. So that's what we'll do. Multiply everything in the bottom equation by -2. When
you do that the set of equations becomes:
.
+ 3x – 4y = 8
-12x + 4y = -20
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Now adding the two equations vertically results in:
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-9x + 0y = -12
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The y term is gone so the equation is:
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-9x = -12
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Divide both sides of this equation by -9 and you find that:
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x = -12/-9 and this reduces to x = 4/3
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You now know what x equals. You can solve for y by returning to either of the original
equations and substituting 4/3 for x. Let's go back to the top equation:
.
3x – 4y = 8
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Substituting 4/3 for x results in:
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3(4/3) - 4y = 8
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Multiply out the left hand term and you get 4. This makes the equation become:
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4 - 4y = 8
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Subtract 4 from both sides to get rid of the 4 on the left side and you get:
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- 4y = 4
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Solve for y by dividing both sides of this equation by -4 and you get:
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y = 4/-4 = -1
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So the answer to this set of equations is x = 4/3 and y = -1
.
Hope this helps you to understand how to solve sets of linear equations by addition.
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
| Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition |
Lets start with the given system of linear equations
In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).
So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.
So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 3 and 6 to some equal number, we could try to get them to the LCM.
Since the LCM of 3 and 6 is 6, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -1 like this:
 Multiply the top equation (both sides) by 2
 Multiply the bottom equation (both sides) by -1
So after multiplying we get this:
Notice how 6 and -6 add to zero (ie  )
Now add the equations together. In order to add 2 equations, group like terms and combine them
 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.
So after adding and canceling out the x terms we're left with:
 Divide both sides by  to solve for y
 Reduce
Now plug this answer into the top equation to solve for x
 Plug in
 Multiply
 Subtract  from both sides
 Combine the terms on the right side
 Multiply both sides by  . This will cancel out  on the left side.
 Multiply the terms on the right side
So our answer is
,
which also looks like
( ,  )
Notice if we graph the equations (if you need help with graphing, check out this solver)
we get
 graph of (red) (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).
and we can see that the two equations intersect at (,). This verifies our answer. |
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