SOLUTION: My teacher doesn't explain well & I don't understand how to do elimination & substitution problems. ex.-3x-1/2y=10 5x+1/4y=8 How do I figure out this problem? Thank you so

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Question 82903: My teacher doesn't explain well & I don't understand how to do elimination & substitution problems.
ex.-3x-1/2y=10
5x+1/4y=8
How do I figure out this problem?
Thank you so much for your help!
Chapter 8.4 Algebra 1
Found 2 solutions by jim_thompson5910, ankor@dixie-net.com:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First multiply the top equation by 2 to eliminate the denominator

2%28-3x-%281%2F2%29y%29=2%2810%29 Multiply both sides by 2

-6x-y=20 Distribute

Now multiply the bottom equation by 4 to eliminate the denominator

4%285x%2B%281%2F4%29y%29=4%288%29 Multiply both sides by 4

20x%2By=32 Distribute

So after multiplying we get the system of equations:
-6x-y=20
20x%2By=32

So if you want to solve by elimination you would follow this procedure

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


Lets start with the given system of linear equations

-6%2Ax-1%2Ay=20
20%2Ax%2B1%2Ay=32

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 -6 and 20 to some equal number, we could try to get them to the LCM.

Since the LCM of -6 and 20 is -60, we need to multiply both sides of the top equation by 10 and multiply both sides of the bottom equation by 3 like this:

10%2A%28-6%2Ax-1%2Ay%29=%2820%29%2A10 Multiply the top equation (both sides) by 10
3%2A%2820%2Ax%2B1%2Ay%29=%2832%29%2A3 Multiply the bottom equation (both sides) by 3


So after multiplying we get this:
-60%2Ax-10%2Ay=200
60%2Ax%2B3%2Ay=96

Notice how -60 and 60 add to zero (ie -60%2B60=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-60%2Ax%2B60%2Ax%29-10%2Ay%2B3%2Ay%29=200%2B96

%28-60%2B60%29%2Ax-10%2B3%29y=200%2B96

cross%28-60%2B60%29%2Ax%2B%28-10%2B3%29%2Ay=200%2B96 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:

-7%2Ay=296

y=296%2F-7 Divide both sides by -7 to solve for y



y=-296%2F7 Reduce


Now plug this answer into the top equation -6%2Ax-1%2Ay=20 to solve for x

-6%2Ax-1%28-296%2F7%29=20 Plug in y=-296%2F7


-6%2Ax%2B296%2F7=20 Multiply



-6%2Ax%2B296%2F7=20 Reduce



-6%2Ax=20-296%2F7 Subtract 296%2F7 from both sides

-6%2Ax=140%2F7-296%2F7 Make 20 into a fraction with a denominator of 7

-6%2Ax=-156%2F7 Combine the terms on the right side

cross%28%281%2F-6%29%28-6%29%29%2Ax=%28-156%2F7%29%281%2F-6%29 Multiply both sides by 1%2F-6. This will cancel out -6 on the left side.


x=26%2F7 Multiply the terms on the right side


So our answer is

x=26%2F7, y=-296%2F7

which also looks like

(26%2F7, -296%2F7)

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

-6%2Ax-1%2Ay=20
20%2Ax%2B1%2Ay=32

we get



graph of -6%2Ax-1%2Ay=20 (red) 20%2Ax%2B1%2Ay=32 (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 (26%2F7,-296%2F7). This verifies our answer.



Or if you want to solve by substitution you would follow this procedure

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

-6%2Ax-1%2Ay=20
20%2Ax%2B1%2Ay=32

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-1%2Ay=20%2B6%2AxAdd 6%2Ax to both sides

y=%2820%2B6%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-20-6%2Ax Now we've fully isolated y

Since y equals -20-6%2Ax we can substitute the expression -20-6%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


20%2Ax%2B1%2Ahighlight%28%28-20-6%2Ax%29%29=32 Replace y with -20-6%2Ax. Since this eliminates y, we can now solve for x.

20%2Ax%2B1%2A%28-20%29%2B1%28-6%29x=32 Distribute 1 to -20-6%2Ax

20%2Ax-20-6%2Ax=32 Multiply



20%2Ax-20-6%2Ax=32 Reduce any fractions

20%2Ax-6%2Ax=32%2B20Add 20 to both sides


20%2Ax-6%2Ax=52 Combine the terms on the right side



14%2Ax=52 Now combine the terms on the left side.


cross%28%281%2F14%29%2814%2F1%29%29x=%2852%2F1%29%281%2F14%29 Multiply both sides by 1%2F14. This will cancel out 14%2F1 and isolate x

So when we multiply 52%2F1 and 1%2F14 (and simplify) we get



x=26%2F7 <---------------------------------One answer

Now that we know that x=26%2F7, lets substitute that in for x to solve for y

20%2826%2F7%29%2B1%2Ay=32 Plug in x=26%2F7 into the 2nd equation

520%2F7%2B1%2Ay=32 Multiply

1%2Ay=32-520%2F7Subtract 520%2F7 from both sides

1%2Ay=224%2F7-520%2F7 Make 32 into a fraction with a denominator of 7



1%2Ay=-296%2F7 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%28-296%2F7%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=-296%2F7 Multiply the terms on the right side


y=-296%2F7 Reduce


So this is the other answer


y=-296%2F7<---------------------------------Other answer


So our solution is

x=26%2F7 and y=-296%2F7

which can also look like

(26%2F7,-296%2F7)

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

-6%2Ax-1%2Ay=20
20%2Ax%2B1%2Ay=32

we get


graph of -6%2Ax-1%2Ay=20 (red) and 20%2Ax%2B1%2Ay=32 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (26%2F7,-296%2F7). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (26%2F7,-296%2F7) into the system of equations


Let x=26%2F7 and y=-296%2F7. Now plug those values into the equation -6%2Ax-1%2Ay=20

-6%2A%2826%2F7%29-1%2A%28-296%2F7%29=20 Plug in x=26%2F7 and y=-296%2F7


-156%2F7%2B296%2F7=20 Multiply


140%2F7=20 Add


20=20 Reduce. Since this equation is true the solution works.


So the solution (26%2F7,-296%2F7) satisfies -6%2Ax-1%2Ay=20



Let x=26%2F7 and y=-296%2F7. Now plug those values into the equation 20%2Ax%2B1%2Ay=32

20%2A%2826%2F7%29%2B1%2A%28-296%2F7%29=32 Plug in x=26%2F7 and y=-296%2F7


520%2F7-296%2F7=32 Multiply


224%2F7=32 Add


32=32 Reduce. Since this equation is true the solution works.


So the solution (26%2F7,-296%2F7) satisfies 20%2Ax%2B1%2Ay=32


Since the solution (26%2F7,-296%2F7) satisfies the system of equations


-6%2Ax-1%2Ay=20
20%2Ax%2B1%2Ay=32


this verifies our answer.

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
how to do elimination & substitution problems.
ex.
:
Use brackets to ensure that you mean y is not part of the denominator;
-3x - (1/2)y = 10
5x + (1/4)y = 8
:
Get rid of the fractions, mult the 1st equation by 2, and the 2nd equation by 4
Resulting in:
-6x - 1y = 20
20x + 1y = 32
:
Notice in this problem, if we just add these two equations we eliminate y:
-6x - y = 20
20x + y = 32
--------------- adding eliminates y, easy to solve for x
14x + 0 = 52
14x = 52
x = 52/14; divide both sides by 14
x = 26/7; lowest terms
:
Pick one of the equations, substitute 3 for x and solve for y:
20x + y = 32
20(26/7) + y = 32
(520/7) + y = 32
y = 32 - (520/7)
y = (224/7) - (520/7); changed 32 to 7ths
y = -296/7
:
Check solutions in one of the original equations:
-3x - (1/2)y = 10
-3(26/7) - (1/2)(-296/7) =
(-78/7) - (-148/7) =
(-78/7) + (148/7) =
+70/7 = 10; proves our solution
:
:
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