SOLUTION: sorry i posted the wrong equation, here are the right ones 2x + y = 7 x + 5y = 12

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Question 178131: sorry i posted the wrong equation, here are the right ones
2x + y = 7
x + 5y = 12
Found 2 solutions by jim_thompson5910, Mathtut:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's solve the system by using substitution:


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


Lets start with the given system of linear equations

2%2Ax%2B1%2Ay=7
1%2Ax%2B5%2Ay=12

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=7-2%2AxSubtract 2%2Ax from both sides

y=%287-2%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=7-2%2Ax Now we've fully isolated y

Since y equals 7-2%2Ax we can substitute the expression 7-2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B5%2Ahighlight%28%287-2%2Ax%29%29=12 Replace y with 7-2%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B5%2A%287%29%2B5%28-2%29x=12 Distribute 5 to 7-2%2Ax

1%2Ax%2B35-10%2Ax=12 Multiply



1%2Ax%2B35-10%2Ax=12 Reduce any fractions

1%2Ax-10%2Ax=12-35 Subtract 35 from both sides


1%2Ax-10%2Ax=-23 Combine the terms on the right side



-9%2Ax=-23 Now combine the terms on the left side.


cross%28%281%2F-9%29%28-9%2F1%29%29x=%28-23%2F1%29%281%2F-9%29 Multiply both sides by 1%2F-9. This will cancel out -9%2F1 and isolate x

So when we multiply -23%2F1 and 1%2F-9 (and simplify) we get



x=23%2F9 <---------------------------------One answer

Now that we know that x=23%2F9, lets substitute that in for x to solve for y

1%2823%2F9%29%2B5%2Ay=12 Plug in x=23%2F9 into the 2nd equation

23%2F9%2B5%2Ay=12 Multiply

5%2Ay=12-23%2F9Subtract 23%2F9 from both sides

5%2Ay=108%2F9-23%2F9 Make 12 into a fraction with a denominator of 9



5%2Ay=85%2F9 Combine the terms on the right side

cross%28%281%2F5%29%285%29%29%2Ay=%2885%2F9%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5 on the left side.

y=85%2F45 Multiply the terms on the right side


y=17%2F9 Reduce


So this is the other answer


y=17%2F9<---------------------------------Other answer


So our solution is

x=23%2F9 and y=17%2F9

which can also look like

(23%2F9,17%2F9)

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

2%2Ax%2B1%2Ay=7
1%2Ax%2B5%2Ay=12

we get


graph of 2%2Ax%2B1%2Ay=7 (red) and 1%2Ax%2B5%2Ay=12 (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 (23%2F9,17%2F9). This verifies our answer.


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Check:

Plug in (23%2F9,17%2F9) into the system of equations


Let x=23%2F9 and y=17%2F9. Now plug those values into the equation 2%2Ax%2B1%2Ay=7

2%2A%2823%2F9%29%2B1%2A%2817%2F9%29=7 Plug in x=23%2F9 and y=17%2F9


46%2F9%2B17%2F9=7 Multiply


63%2F9=7 Add


7=7 Reduce. Since this equation is true the solution works.


So the solution (23%2F9,17%2F9) satisfies 2%2Ax%2B1%2Ay=7



Let x=23%2F9 and y=17%2F9. Now plug those values into the equation 1%2Ax%2B5%2Ay=12

1%2A%2823%2F9%29%2B5%2A%2817%2F9%29=12 Plug in x=23%2F9 and y=17%2F9


23%2F9%2B85%2F9=12 Multiply


108%2F9=12 Add


12=12 Reduce. Since this equation is true the solution works.


So the solution (23%2F9,17%2F9) satisfies 1%2Ax%2B5%2Ay=12


Since the solution (23%2F9,17%2F9) satisfies the system of equations


2%2Ax%2B1%2Ay=7
1%2Ax%2B5%2Ay=12


this verifies our answer.

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
substitution: take one equation and isolate a variable then take the value of the variable and plug it into the OTHER equation
:
2x+y=7.....eq 1
x+5y=12....eq 2
:
rewrite eq 1 to y=-2x+7....now take the value of y which is -2x+7 and plug that into eq 2 for every y that you see
:
x+5(-2x+7)=12
:
x-10x+35=12.......distributed left side
:
-9x=-23............combined like terms on either side
:
highlight%28x=23%2F9%29
:
now take the found value of x and plug it into either equation to find y
:
y=-2x+7--->y=-2(23/9)+7--->-46/9+63/9=highlight%2817%2F9%29
:
Elimination or addition method: This is manipulating one or both of the equations so as to eliminate one of the variables. It is really important, when learning this method to always line up the terms so as to SEE what is taking place
:.
2x+y=7.....eq 1
x+5y=12....eq 2
:
We can either choose to eliminate the x or y terms. To eliminate the y terms we would multiply eq 1 by -5 and add the equations together or we can eliminate the x terms by multiplying eq 2 by -2 and adding the equations together. I choose to eliminate the x terms
:
eq 2----->-2(x+5y=12)---->-2x-10y=-24....now take this and line it up under eq 1
:
2x+y=7........eq 1
-2x-10y=-24...eq 2 revised
:
now you can clearly see if you add the terms of each equation together that the x terms are eliminated because 2x-2x=0. We are left with y-10y=7-24
:
-9y=-17
:
highlight%28y=17%2F9%29
:
now take y's found value and plug it back into any numbered equation we have.
I choose eq 1
:
2x+17/9=7......plugged in y value of 17/9
:
18x+17=63......multiplied each term by 9 to get rid of fraction
:
18x=46
:
highlight%28x=46%2F18=23%2F9%29
:
w%5E5which was what we wanted......
:
Mathtut