SOLUTION: Solve each of the following by subtraction. 5x – 2y = -5 y – 5x = 3 8x – 4y = 16 y = 2x – 4 4x – 12y = 5 -x + 3y = -1

Algebra ->  Expressions-with-variables -> SOLUTION: Solve each of the following by subtraction. 5x – 2y = -5 y – 5x = 3 8x – 4y = 16 y = 2x – 4 4x – 12y = 5 -x + 3y = -1       Log On


   

Question 85366: Solve each of the following by subtraction.
5x – 2y = -5 y – 5x = 3
8x – 4y = 16 y = 2x – 4
4x – 12y = 5 -x + 3y = -1
10x + 2y = 7 y = -5x + 3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that the problems are divided into columns
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

5%2Ax-2%2Ay=-5
8%2Ax-4%2Ay=16

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 8 to some equal number, we could try to get them to the LCM.

Since the LCM of 5 and 8 is 40, we need to multiply both sides of the top equation by 8 and multiply both sides of the bottom equation by -5 like this:

8%2A%285%2Ax-2%2Ay%29=%28-5%29%2A8 Multiply the top equation (both sides) by 8
-5%2A%288%2Ax-4%2Ay%29=%2816%29%2A-5 Multiply the bottom equation (both sides) by -5


So after multiplying we get this:
40%2Ax-16%2Ay=-40
-40%2Ax%2B20%2Ay=-80

Notice how 40 and -40 add to zero (ie 40%2B-40=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2840%2Ax-40%2Ax%29-16%2Ay%2B20%2Ay%29=-40-80

%2840-40%29%2Ax-16%2B20%29y=-40-80

cross%2840%2B-40%29%2Ax%2B%28-16%2B20%29%2Ay=-40-80 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:

4%2Ay=-120

y=-120%2F4 Divide both sides by 4 to solve for y



y=-30 Reduce


Now plug this answer into the top equation 5%2Ax-2%2Ay=-5 to solve for x

5%2Ax-2%28-30%29=-5 Plug in y=-30


5%2Ax%2B60=-5 Multiply



5%2Ax=-5-60 Subtract 60 from both sides

5%2Ax=-65 Combine the terms on the right side

cross%28%281%2F5%29%285%29%29%2Ax=%28-65%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5 on the left side.


x=-13 Multiply the terms on the right side


So our answer is

x=-13, y=-30

which also looks like

(-13, -30)

Notice if we graph the equations (if you need help with graphing, check out this solver)

5%2Ax-2%2Ay=-5
8%2Ax-4%2Ay=16

we get



graph of 5%2Ax-2%2Ay=-5 (red) 8%2Ax-4%2Ay=16 (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 (-13,-30). This verifies our answer.



Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

4%2Ax-12%2Ay=5
10%2Ax%2B2%2Ay=7

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 4 and 10 to some equal number, we could try to get them to the LCM.

Since the LCM of 4 and 10 is 20, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -2 like this:

5%2A%284%2Ax-12%2Ay%29=%285%29%2A5 Multiply the top equation (both sides) by 5
-2%2A%2810%2Ax%2B2%2Ay%29=%287%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
20%2Ax-60%2Ay=25
-20%2Ax-4%2Ay=-14

Notice how 20 and -20 add to zero (ie 20%2B-20=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2820%2Ax-20%2Ax%29-60%2Ay-4%2Ay%29=25-14

%2820-20%29%2Ax-60-4%29y=25-14

cross%2820%2B-20%29%2Ax%2B%28-60-4%29%2Ay=25-14 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:

-64%2Ay=11

y=11%2F-64 Divide both sides by -64 to solve for y



y=-11%2F64 Reduce


Now plug this answer into the top equation 4%2Ax-12%2Ay=5 to solve for x

4%2Ax-12%28-11%2F64%29=5 Plug in y=-11%2F64


4%2Ax%2B132%2F64=5 Multiply



4%2Ax%2B33%2F16=5 Reduce



4%2Ax=5-33%2F16 Subtract 33%2F16 from both sides

4%2Ax=80%2F16-33%2F16 Make 5 into a fraction with a denominator of 16

4%2Ax=47%2F16 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ax=%2847%2F16%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.


x=47%2F64 Multiply the terms on the right side


So our answer is

x=47%2F64, y=-11%2F64

which also looks like

(47%2F64, -11%2F64)

Notice if we graph the equations (if you need help with graphing, check out this solver)

4%2Ax-12%2Ay=5
10%2Ax%2B2%2Ay=7

we get



graph of 4%2Ax-12%2Ay=5 (red) 10%2Ax%2B2%2Ay=7 (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 (47%2F64,-11%2F64). This verifies our answer.



y+%96+5x+=+3
y+=+2x+%96+4+

y+%96+5x+=+3
y+-+2x+=+4+ Subtract 2x from both sides
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

-5%2Ax%2B1%2Ay=3
-2%2Ax%2B1%2Ay=-4

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 -2 to some equal number, we could try to get them to the LCM.

Since the LCM of -5 and -2 is 10, we need to multiply both sides of the top equation by -2 and multiply both sides of the bottom equation by 5 like this:

-2%2A%28-5%2Ax%2B1%2Ay%29=%283%29%2A-2 Multiply the top equation (both sides) by -2
5%2A%28-2%2Ax%2B1%2Ay%29=%28-4%29%2A5 Multiply the bottom equation (both sides) by 5


So after multiplying we get this:
10%2Ax-2%2Ay=-6
-10%2Ax%2B5%2Ay=-20

Notice how 10 and -10 add to zero (ie 10%2B-10=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2810%2Ax-10%2Ax%29-2%2Ay%2B5%2Ay%29=-6-20

%2810-10%29%2Ax-2%2B5%29y=-6-20

cross%2810%2B-10%29%2Ax%2B%28-2%2B5%29%2Ay=-6-20 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:

3%2Ay=-26

y=-26%2F3 Divide both sides by 3 to solve for y



y=-26%2F3 Reduce


Now plug this answer into the top equation -5%2Ax%2B1%2Ay=3 to solve for x

-5%2Ax%2B1%28-26%2F3%29=3 Plug in y=-26%2F3


-5%2Ax-26%2F3=3 Multiply



-5%2Ax-26%2F3=3 Reduce



-5%2Ax=3%2B26%2F3 Subtract -26%2F3 from both sides

-5%2Ax=9%2F3%2B26%2F3 Make 3 into a fraction with a denominator of 3

-5%2Ax=35%2F3 Combine the terms on the right side

cross%28%281%2F-5%29%28-5%29%29%2Ax=%2835%2F3%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5 on the left side.


x=-7%2F3 Multiply the terms on the right side


So our answer is

x=-7%2F3, y=-26%2F3

which also looks like

(-7%2F3, -26%2F3)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-5%2Ax%2B1%2Ay=3
-2%2Ax%2B1%2Ay=-4

we get



graph of -5%2Ax%2B1%2Ay=3 (red) -2%2Ax%2B1%2Ay=-4 (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 (-7%2F3,-26%2F3). This verifies our answer.



-x+%2B+3y+=+-1
y+=+-5x+%2B+3+


-x+%2B+3y+=+-1
y%2B5x+=+3+ Add 5x to both sides
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

-1%2Ax%2B3%2Ay=-1
5%2Ax%2B1%2Ay=3

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 5 to some equal number, we could try to get them to the LCM.

Since the LCM of -1 and 5 is -5, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by 1 like this:

5%2A%28-1%2Ax%2B3%2Ay%29=%28-1%29%2A5 Multiply the top equation (both sides) by 5
1%2A%285%2Ax%2B1%2Ay%29=%283%29%2A1 Multiply the bottom equation (both sides) by 1


So after multiplying we get this:
-5%2Ax%2B15%2Ay=-5
5%2Ax%2B1%2Ay=3

Notice how -5 and 5 add to zero (ie -5%2B5=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-5%2Ax%2B5%2Ax%29%2B%2815%2Ay%2B1%2Ay%29=-5%2B3

%28-5%2B5%29%2Ax%2B%2815%2B1%29y=-5%2B3

cross%28-5%2B5%29%2Ax%2B%2815%2B1%29%2Ay=-5%2B3 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:

16%2Ay=-2

y=-2%2F16 Divide both sides by 16 to solve for y



y=-1%2F8 Reduce


Now plug this answer into the top equation -1%2Ax%2B3%2Ay=-1 to solve for x

-1%2Ax%2B3%28-1%2F8%29=-1 Plug in y=-1%2F8


-1%2Ax-3%2F8=-1 Multiply



-1%2Ax-3%2F8=-1 Reduce



-1%2Ax=-1%2B3%2F8 Subtract -3%2F8 from both sides

-1%2Ax=-8%2F8%2B3%2F8 Make -1 into a fraction with a denominator of 8

-1%2Ax=-5%2F8 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ax=%28-5%2F8%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.


x=5%2F8 Multiply the terms on the right side


So our answer is

x=5%2F8, y=-1%2F8

which also looks like

(5%2F8, -1%2F8)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-1%2Ax%2B3%2Ay=-1
5%2Ax%2B1%2Ay=3

we get



graph of -1%2Ax%2B3%2Ay=-1 (red) 5%2Ax%2B1%2Ay=3 (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 (5%2F8,-1%2F8). This verifies our answer.