SOLUTION: Solve each system by substitution 4y-2x=-2 and x+3y=4

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Question 701810: Solve each system by substitution
4y-2x=-2 and x+3y=4
Answer by mouk(232) 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

1%2Ax%2B3%2Ay=4
-2%2Ax%2B4%2Ay=-2

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

3%2Ay=4-1%2AxSubtract 1%2Ax from both sides

y=%284-1%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=4%2F3-%281%2F3%29%2Ax Now we've fully isolated y

Since y equals 4%2F3-%281%2F3%29%2Ax we can substitute the expression 4%2F3-%281%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-2%2Ax%2B4%2Ahighlight%28%284%2F3-%281%2F3%29%2Ax%29%29=-2 Replace y with 4%2F3-%281%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

-2%2Ax%2B4%2A%284%2F3%29%2B4%28-1%2F3%29x=-2 Distribute 4 to 4%2F3-%281%2F3%29%2Ax

-2%2Ax%2B16%2F3-%284%2F3%29%2Ax=-2 Multiply



-2%2Ax%2B16%2F3-%284%2F3%29%2Ax=-2 Reduce any fractions

-2%2Ax-%284%2F3%29%2Ax=-2-16%2F3 Subtract 16%2F3 from both sides


-2%2Ax-%284%2F3%29%2Ax=-6%2F3-16%2F3 Make -2 into a fraction with a denominator of 3


-2%2Ax-%284%2F3%29%2Ax=-22%2F3 Combine the terms on the right side



%28-6%2F3%29%2Ax-%284%2F3%29x=-22%2F3 Make -2 into a fraction with a denominator of 3

%28-10%2F3%29%2Ax=-22%2F3 Now combine the terms on the left side.


cross%28%283%2F-10%29%28-10%2F3%29%29x=%28-22%2F3%29%283%2F-10%29 Multiply both sides by 3%2F-10. This will cancel out -10%2F3 and isolate x

So when we multiply -22%2F3 and 3%2F-10 (and simplify) we get



x=11%2F5 <---------------------------------One answer

Now that we know that x=11%2F5, lets substitute that in for x to solve for y

-2%2811%2F5%29%2B4%2Ay=-2 Plug in x=11%2F5 into the 2nd equation

-22%2F5%2B4%2Ay=-2 Multiply

4%2Ay=-2%2B22%2F5Add 22%2F5 to both sides

4%2Ay=-10%2F5%2B22%2F5 Make -2 into a fraction with a denominator of 5



4%2Ay=12%2F5 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ay=%2812%2F5%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.

y=12%2F20 Multiply the terms on the right side


y=3%2F5 Reduce


So this is the other answer


y=3%2F5<---------------------------------Other answer


So our solution is

x=11%2F5 and y=3%2F5

which can also look like

(11%2F5,3%2F5)

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

1%2Ax%2B3%2Ay=4
-2%2Ax%2B4%2Ay=-2

we get


graph of 1%2Ax%2B3%2Ay=4 (red) and -2%2Ax%2B4%2Ay=-2 (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 (11%2F5,3%2F5). This verifies our answer.


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Check:

Plug in (11%2F5,3%2F5) into the system of equations


Let x=11%2F5 and y=3%2F5. Now plug those values into the equation 1%2Ax%2B3%2Ay=4

1%2A%2811%2F5%29%2B3%2A%283%2F5%29=4 Plug in x=11%2F5 and y=3%2F5


11%2F5%2B9%2F5=4 Multiply


20%2F5=4 Add


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


So the solution (11%2F5,3%2F5) satisfies 1%2Ax%2B3%2Ay=4



Let x=11%2F5 and y=3%2F5. Now plug those values into the equation -2%2Ax%2B4%2Ay=-2

-2%2A%2811%2F5%29%2B4%2A%283%2F5%29=-2 Plug in x=11%2F5 and y=3%2F5


-22%2F5%2B12%2F5=-2 Multiply


-10%2F5=-2 Add


-2=-2 Reduce. Since this equation is true the solution works.


So the solution (11%2F5,3%2F5) satisfies -2%2Ax%2B4%2Ay=-2


Since the solution (11%2F5,3%2F5) satisfies the system of equations


1%2Ax%2B3%2Ay=4
-2%2Ax%2B4%2Ay=-2


this verifies our answer.