SOLUTION: i need help sloving this system if equation by substitution 2x-2y=4 x+3y=1

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: i need help sloving this system if equation by substitution 2x-2y=4 x+3y=1      Log On


   

Question 120015This question is from textbook
: i need help sloving this system if equation by substitution
2x-2y=4
x+3y=1 This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

2%2Ax-2%2Ay=4
1%2Ax%2B3%2Ay=1

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

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

y=%284-2%2Ax%29%2F-2 Divide both sides by -2.


Which breaks down and reduces to



y=-2%2B1%2Ax Now we've fully isolated y

Since y equals -2%2B1%2Ax we can substitute the expression -2%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B3%2Ahighlight%28%28-2%2B1%2Ax%29%29=1 Replace y with -2%2B1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B3%2A%28-2%29%2B3%281%29x=1 Distribute 3 to -2%2B1%2Ax

1%2Ax-6%2B3%2Ax=1 Multiply



1%2Ax-6%2B3%2Ax=1 Reduce any fractions

1%2Ax%2B3%2Ax=1%2B6Add 6 to both sides


1%2Ax%2B3%2Ax=7 Combine the terms on the right side



4%2Ax=7 Now combine the terms on the left side.


cross%28%281%2F4%29%284%2F1%29%29x=%287%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4%2F1 and isolate x

So when we multiply 7%2F1 and 1%2F4 (and simplify) we get



x=7%2F4 <---------------------------------One answer

Now that we know that x=7%2F4, lets substitute that in for x to solve for y

1%287%2F4%29%2B3%2Ay=1 Plug in x=7%2F4 into the 2nd equation

7%2F4%2B3%2Ay=1 Multiply

3%2Ay=1-7%2F4Subtract 7%2F4 from both sides

3%2Ay=4%2F4-7%2F4 Make 1 into a fraction with a denominator of 4



3%2Ay=-3%2F4 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ay=%28-3%2F4%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.

y=-3%2F12 Multiply the terms on the right side


y=-1%2F4 Reduce


So this is the other answer


y=-1%2F4<---------------------------------Other answer


So our solution is

x=7%2F4 and y=-1%2F4

which can also look like

(7%2F4,-1%2F4)

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

2%2Ax-2%2Ay=4
1%2Ax%2B3%2Ay=1

we get


graph of 2%2Ax-2%2Ay=4 (red) and 1%2Ax%2B3%2Ay=1 (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 (7%2F4,-1%2F4). This verifies our answer.


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

Plug in (7%2F4,-1%2F4) into the system of equations


Let x=7%2F4 and y=-1%2F4. Now plug those values into the equation 2%2Ax-2%2Ay=4

2%2A%287%2F4%29-2%2A%28-1%2F4%29=4 Plug in x=7%2F4 and y=-1%2F4


14%2F4%2B2%2F4=4 Multiply


16%2F4=4 Add


4=4 Reduce. Since this equation is true the solution works.


So the solution (7%2F4,-1%2F4) satisfies 2%2Ax-2%2Ay=4



Let x=7%2F4 and y=-1%2F4. Now plug those values into the equation 1%2Ax%2B3%2Ay=1

1%2A%287%2F4%29%2B3%2A%28-1%2F4%29=1 Plug in x=7%2F4 and y=-1%2F4


7%2F4-3%2F4=1 Multiply


4%2F4=1 Add


1=1 Reduce. Since this equation is true the solution works.


So the solution (7%2F4,-1%2F4) satisfies 1%2Ax%2B3%2Ay=1


Since the solution (7%2F4,-1%2F4) satisfies the system of equations


2%2Ax-2%2Ay=4
1%2Ax%2B3%2Ay=1


this verifies our answer.