SOLUTION: 5x-4y=1 3x-6y=6

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Question 115285: 5x-4y=1
3x-6y=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

5%2Ax-4%2Ay=1
3%2Ax-6%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=1-5%2AxSubtract 5%2Ax from both sides

y=%281-5%2Ax%29%2F-4 Divide both sides by -4.


Which breaks down and reduces to



y=-1%2F4%2B%285%2F4%29%2Ax Now we've fully isolated y

Since y equals -1%2F4%2B%285%2F4%29%2Ax we can substitute the expression -1%2F4%2B%285%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-6%2Ahighlight%28%28-1%2F4%2B%285%2F4%29%2Ax%29%29=6 Replace y with -1%2F4%2B%285%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-6%2A%28-1%2F4%29-6%285%2F4%29x=6 Distribute -6 to -1%2F4%2B%285%2F4%29%2Ax

3%2Ax%2B6%2F4-%2830%2F4%29%2Ax=6 Multiply



3%2Ax%2B3%2F2-%2815%2F2%29%2Ax=6 Reduce any fractions

3%2Ax-%2815%2F2%29%2Ax=6-3%2F2 Subtract 3%2F2 from both sides


3%2Ax-%2815%2F2%29%2Ax=12%2F2-3%2F2 Make 6 into a fraction with a denominator of 2


3%2Ax-%2815%2F2%29%2Ax=9%2F2 Combine the terms on the right side



%286%2F2%29%2Ax-%2815%2F2%29x=9%2F2 Make 3 into a fraction with a denominator of 2

%28-9%2F2%29%2Ax=9%2F2 Now combine the terms on the left side.


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

So when we multiply 9%2F2 and 2%2F-9 (and simplify) we get



x=-1 <---------------------------------One answer

Now that we know that x=-1, lets substitute that in for x to solve for y

3%28-1%29-6%2Ay=6 Plug in x=-1 into the 2nd equation

-3-6%2Ay=6 Multiply

-6%2Ay=6%2B3Add 3 to both sides

-6%2Ay=9 Combine the terms on the right side

cross%28%281%2F-6%29%28-6%29%29%2Ay=%289%2F1%29%281%2F-6%29 Multiply both sides by 1%2F-6. This will cancel out -6 on the left side.

y=9%2F-6 Multiply the terms on the right side


y=-3%2F2 Reduce


So this is the other answer


y=-3%2F2<---------------------------------Other answer


So our solution is

x=-1 and y=-3%2F2

which can also look like

(-1,-3%2F2)

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

5%2Ax-4%2Ay=1
3%2Ax-6%2Ay=6

we get


graph of 5%2Ax-4%2Ay=1 (red) and 3%2Ax-6%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 (-1,-3%2F2). This verifies our answer.


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Check:

Plug in (-1,-3%2F2) into the system of equations


Let x=-1 and y=-3%2F2. Now plug those values into the equation 5%2Ax-4%2Ay=1

5%2A%28-1%29-4%2A%28-3%2F2%29=1 Plug in x=-1 and y=-3%2F2


-5%2B12%2F2=1 Multiply


2%2F2=1 Add


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


So the solution (-1,-3%2F2) satisfies 5%2Ax-4%2Ay=1



Let x=-1 and y=-3%2F2. Now plug those values into the equation 3%2Ax-6%2Ay=6

3%2A%28-1%29-6%2A%28-3%2F2%29=6 Plug in x=-1 and y=-3%2F2


-3%2B18%2F2=6 Multiply


12%2F2=6 Add


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


So the solution (-1,-3%2F2) satisfies 3%2Ax-6%2Ay=6


Since the solution (-1,-3%2F2) satisfies the system of equations


5%2Ax-4%2Ay=1
3%2Ax-6%2Ay=6


this verifies our answer.