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

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Question 115282: 3x+4y=5
x-2x=-5
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%2B4%2Ay=5
1%2Ax-2%2Ay=-5

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%2F4 Divide both sides by 4.


Which breaks down and reduces to



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

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


1%2Ax%2B-2%2Ahighlight%28%285%2F4-%283%2F4%29%2Ax%29%29=-5 Replace y with 5%2F4-%283%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-2%2A%285%2F4%29-2%28-3%2F4%29x=-5 Distribute -2 to 5%2F4-%283%2F4%29%2Ax

1%2Ax-10%2F4%2B%286%2F4%29%2Ax=-5 Multiply



1%2Ax-5%2F2%2B%283%2F2%29%2Ax=-5 Reduce any fractions

1%2Ax%2B%283%2F2%29%2Ax=-5%2B5%2F2Add 5%2F2 to both sides


1%2Ax%2B%283%2F2%29%2Ax=-10%2F2%2B5%2F2 Make -5 into a fraction with a denominator of 2


1%2Ax%2B%283%2F2%29%2Ax=-5%2F2 Combine the terms on the right side



%282%2F2%29%2Ax%2B%283%2F2%29x=-5%2F2 Make 1 into a fraction with a denominator of 2

%285%2F2%29%2Ax=-5%2F2 Now combine the terms on the left side.


cross%28%282%2F5%29%285%2F2%29%29x=%28-5%2F2%29%282%2F5%29 Multiply both sides by 2%2F5. This will cancel out 5%2F2 and isolate x

So when we multiply -5%2F2 and 2%2F5 (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

1%28-1%29-2%2Ay=-5 Plug in x=-1 into the 2nd equation

-1-2%2Ay=-5 Multiply

-2%2Ay=-5%2B1Add 1 to both sides

-2%2Ay=-4 Combine the terms on the right side

cross%28%281%2F-2%29%28-2%29%29%2Ay=%28-4%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2 on the left side.

y=-4%2F-2 Multiply the terms on the right side


y=2 Reduce


So this is the other answer


y=2<---------------------------------Other answer


So our solution is

x=-1 and y=2

which can also look like

(-1,2)

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

3%2Ax%2B4%2Ay=5
1%2Ax-2%2Ay=-5

we get


graph of 3%2Ax%2B4%2Ay=5 (red) and 1%2Ax-2%2Ay=-5 (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 (-1,2). This verifies our answer.


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Check:

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


Let x=-1 and y=2. Now plug those values into the equation 3%2Ax%2B4%2Ay=5

3%2A%28-1%29%2B4%2A%282%29=5 Plug in x=-1 and y=2


-3%2B8=5 Multiply


5=5 Add


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


So the solution (-1,2) satisfies 3%2Ax%2B4%2Ay=5



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

1%2A%28-1%29-2%2A%282%29=-5 Plug in x=-1 and y=2


-1-4=-5 Multiply


-5=-5 Add


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


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


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


3%2Ax%2B4%2Ay=5
1%2Ax-2%2Ay=-5


this verifies our answer.