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SOLUTION: solve the system by addition, x+y=2 x-y=4 5x-3y=13 4x-3y=11
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Question 82595
:
solve the system by addition,
x+y=2
x-y=4
5x-3y=13
4x-3y=11
Answer by
jim_thompson5910(33401)
(
Show Source
):
You can
put this solution on YOUR website!
Lets start with
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 1 and 1 to some equal number, we could try to get them to the LCM.
Since the LCM of 1 and 1 is 1, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -1 like this:
Multiply the top equation (both sides) by 1
Multiply the bottom equation (both sides) by -1
So after multiplying we get this:
Notice how 1 and -1 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.
Now lets solve
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 5 and 4 to some equal number, we could try to get them to the LCM.
Since the LCM of 5 and 4 is 20, we need to multiply both sides of the top equation by 4 and multiply both sides of the bottom equation by -5 like this:
Multiply the top equation (both sides) by 4
Multiply the bottom equation (both sides) by -5
So after multiplying we get this:
Notice how 20 and -20 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
Reduce
Subtract
from both sides
Make 13 into a fraction with a denominator of 3
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.