SOLUTION: 3x-4y=5 -4x+2y=6

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Question 118519: 3x-4y=5
-4x+2y=6
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

3%2Ax-4%2Ay=5
-4%2Ax%2B2%2Ay=6

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=5-3%2AxSubtract 3%2Ax from both sides

y=%285-3%2Ax%29%2F-4 Divide both sides by -4.


Which breaks down and reduces to



y=-5%2F4%2B%283%2F4%29%2Ax Now we've fully isolated y

Since y equals -5%2F4%2B%283%2F4%29%2Ax we can substitute the expression -5%2F4%2B%283%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-4%2Ax%2B2%2Ahighlight%28%28-5%2F4%2B%283%2F4%29%2Ax%29%29=6 Replace y with -5%2F4%2B%283%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

-4%2Ax%2B2%2A%28-5%2F4%29%2B2%283%2F4%29x=6 Distribute 2 to -5%2F4%2B%283%2F4%29%2Ax

-4%2Ax-10%2F4%2B%286%2F4%29%2Ax=6 Multiply



-4%2Ax-5%2F2%2B%283%2F2%29%2Ax=6 Reduce any fractions

-4%2Ax%2B%283%2F2%29%2Ax=6%2B5%2F2Add 5%2F2 to both sides


-4%2Ax%2B%283%2F2%29%2Ax=12%2F2%2B5%2F2 Make 6 into a fraction with a denominator of 2


-4%2Ax%2B%283%2F2%29%2Ax=17%2F2 Combine the terms on the right side



%28-8%2F2%29%2Ax%2B%283%2F2%29x=17%2F2 Make -4 into a fraction with a denominator of 2

%28-5%2F2%29%2Ax=17%2F2 Now combine the terms on the left side.


cross%28%282%2F-5%29%28-5%2F2%29%29x=%2817%2F2%29%282%2F-5%29 Multiply both sides by 2%2F-5. This will cancel out -5%2F2 and isolate x

So when we multiply 17%2F2 and 2%2F-5 (and simplify) we get



x=-17%2F5 <---------------------------------One answer

Now that we know that x=-17%2F5, lets substitute that in for x to solve for y

-4%28-17%2F5%29%2B2%2Ay=6 Plug in x=-17%2F5 into the 2nd equation

68%2F5%2B2%2Ay=6 Multiply

2%2Ay=6-68%2F5Subtract 68%2F5 from both sides

2%2Ay=30%2F5-68%2F5 Make 6 into a fraction with a denominator of 5



2%2Ay=-38%2F5 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ay=%28-38%2F5%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.

y=-38%2F10 Multiply the terms on the right side


y=-19%2F5 Reduce


So this is the other answer


y=-19%2F5<---------------------------------Other answer


So our solution is

x=-17%2F5 and y=-19%2F5

which can also look like

(-17%2F5,-19%2F5)

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

3%2Ax-4%2Ay=5
-4%2Ax%2B2%2Ay=6

we get


graph of 3%2Ax-4%2Ay=5 (red) and -4%2Ax%2B2%2Ay=6 (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 (-17%2F5,-19%2F5). This verifies our answer.


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Check:

Plug in (-17%2F5,-19%2F5) into the system of equations


Let x=-17%2F5 and y=-19%2F5. Now plug those values into the equation 3%2Ax-4%2Ay=5

3%2A%28-17%2F5%29-4%2A%28-19%2F5%29=5 Plug in x=-17%2F5 and y=-19%2F5


-51%2F5%2B76%2F5=5 Multiply


25%2F5=5 Add


5=5 Reduce. Since this equation is true the solution works.


So the solution (-17%2F5,-19%2F5) satisfies 3%2Ax-4%2Ay=5



Let x=-17%2F5 and y=-19%2F5. Now plug those values into the equation -4%2Ax%2B2%2Ay=6

-4%2A%28-17%2F5%29%2B2%2A%28-19%2F5%29=6 Plug in x=-17%2F5 and y=-19%2F5


68%2F5-38%2F5=6 Multiply


30%2F5=6 Add


6=6 Reduce. Since this equation is true the solution works.


So the solution (-17%2F5,-19%2F5) satisfies -4%2Ax%2B2%2Ay=6


Since the solution (-17%2F5,-19%2F5) satisfies the system of equations


3%2Ax-4%2Ay=5
-4%2Ax%2B2%2Ay=6


this verifies our answer.