SOLUTION: Solve the system by substitution. x + y = 12 y = 2x Solve the system by addition or substitution. 3x – 4y = 8 6x – 2y = 10

Algebra ->  Linear-equations -> SOLUTION: Solve the system by substitution. x + y = 12 y = 2x Solve the system by addition or substitution. 3x – 4y = 8 6x – 2y = 10       Log On


   

Question 89302: Solve the system by substitution. x + y = 12 y = 2x
Solve the system by addition or substitution. 3x – 4y = 8 6x – 2y = 10
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"Solve the system by substitution. x + y = 12 y = 2x"

x+%2B+y+=+12
x+%2B+highlight%282x%29+=+12 Plug in y=2x (ie replace y with 2x)

3x=12 Combine like terms

x=4 Divide both sides by 3

4+%2B+y+=+12 Plug in x=4

y+=+8 Subtract 4 from both sides

So our answer is (4,8)


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"Solve the system by addition or substitution. 3x – 4y = 8 6x – 2y = 10"

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


Lets start with the given system of linear equations

3%2Ax-4%2Ay=8
6%2Ax-2%2Ay=10

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

-4%2Ay=8-3%2AxSubtract 3%2Ax from both sides

y=%288-3%2Ax%29%2F-4 Divide both sides by -4.


Which breaks down and reduces to



y=-2%2B%283%2F4%29%2Ax Now we've fully isolated y

Since y equals -2%2B%283%2F4%29%2Ax we can substitute the expression -2%2B%283%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


6%2Ax%2B-2%2Ahighlight%28%28-2%2B%283%2F4%29%2Ax%29%29=10 Replace y with -2%2B%283%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

6%2Ax-2%2A%28-2%29-2%283%2F4%29x=10 Distribute -2 to -2%2B%283%2F4%29%2Ax

6%2Ax%2B4-%286%2F4%29%2Ax=10 Multiply



6%2Ax%2B4-%283%2F2%29%2Ax=10 Reduce any fractions

6%2Ax-%283%2F2%29%2Ax=10-4 Subtract 4 from both sides


6%2Ax-%283%2F2%29%2Ax=6 Combine the terms on the right side



%2812%2F2%29%2Ax-%283%2F2%29x=6 Make 6 into a fraction with a denominator of 2

%289%2F2%29%2Ax=6 Now combine the terms on the left side.


cross%28%282%2F9%29%289%2F2%29%29x=%286%2F1%29%282%2F9%29 Multiply both sides by 2%2F9. This will cancel out 9%2F2 and isolate x

So when we multiply 6%2F1 and 2%2F9 (and simplify) we get



x=4%2F3 <---------------------------------One answer

Now that we know that x=4%2F3, lets substitute that in for x to solve for y

6%284%2F3%29-2%2Ay=10 Plug in x=4%2F3 into the 2nd equation

8-2%2Ay=10 Multiply

-2%2Ay=10-8Subtract 8 from both sides

-2%2Ay=2 Combine the terms on the right side

cross%28%281%2F-2%29%28-2%29%29%2Ay=%282%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2 on the left side.

y=2%2F-2 Multiply the terms on the right side


y=-1 Reduce


So this is the other answer


y=-1<---------------------------------Other answer


So our solution is

x=4%2F3 and y=-1

which can also look like

(4%2F3,-1)

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

3%2Ax-4%2Ay=8
6%2Ax-2%2Ay=10

we get


graph of 3%2Ax-4%2Ay=8 (red) and 6%2Ax-2%2Ay=10 (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 (4%2F3,-1). This verifies our answer.


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Check:

Plug in (4%2F3,-1) into the system of equations


Let x=4%2F3 and y=-1. Now plug those values into the equation 3%2Ax-4%2Ay=8

3%2A%284%2F3%29-4%2A%28-1%29=8 Plug in x=4%2F3 and y=-1


12%2F3%2B4=8 Multiply


24%2F3=8 Add


8=8 Reduce. Since this equation is true the solution works.


So the solution (4%2F3,-1) satisfies 3%2Ax-4%2Ay=8



Let x=4%2F3 and y=-1. Now plug those values into the equation 6%2Ax-2%2Ay=10

6%2A%284%2F3%29-2%2A%28-1%29=10 Plug in x=4%2F3 and y=-1


24%2F3%2B2=10 Multiply


30%2F3=10 Add


10=10 Reduce. Since this equation is true the solution works.


So the solution (4%2F3,-1) satisfies 6%2Ax-2%2Ay=10


Since the solution (4%2F3,-1) satisfies the system of equations


3%2Ax-4%2Ay=8
6%2Ax-2%2Ay=10


this verifies our answer.