SOLUTION: 2x-5y=10 4x+3y=7

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Question 114022: 2x-5y=10
4x+3y=7
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-5%2Ay=10
4%2Ax%2B3%2Ay=7

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

-5%2Ay=10-2%2AxSubtract 2%2Ax from both sides

y=%2810-2%2Ax%29%2F-5 Divide both sides by -5.


Which breaks down and reduces to



y=-2%2B%282%2F5%29%2Ax Now we've fully isolated y

Since y equals -2%2B%282%2F5%29%2Ax we can substitute the expression -2%2B%282%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B3%2Ahighlight%28%28-2%2B%282%2F5%29%2Ax%29%29=7 Replace y with -2%2B%282%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax%2B3%2A%28-2%29%2B3%282%2F5%29x=7 Distribute 3 to -2%2B%282%2F5%29%2Ax

4%2Ax-6%2B%286%2F5%29%2Ax=7 Multiply



4%2Ax-6%2B%286%2F5%29%2Ax=7 Reduce any fractions

4%2Ax%2B%286%2F5%29%2Ax=7%2B6Add 6 to both sides


4%2Ax%2B%286%2F5%29%2Ax=13 Combine the terms on the right side



%2820%2F5%29%2Ax%2B%286%2F5%29x=13 Make 4 into a fraction with a denominator of 5

%2826%2F5%29%2Ax=13 Now combine the terms on the left side.


cross%28%285%2F26%29%2826%2F5%29%29x=%2813%2F1%29%285%2F26%29 Multiply both sides by 5%2F26. This will cancel out 26%2F5 and isolate x

So when we multiply 13%2F1 and 5%2F26 (and simplify) we get



x=5%2F2 <---------------------------------One answer

Now that we know that x=5%2F2, lets substitute that in for x to solve for y

4%285%2F2%29%2B3%2Ay=7 Plug in x=5%2F2 into the 2nd equation

10%2B3%2Ay=7 Multiply

3%2Ay=7-10Subtract 10 from both sides

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

cross%28%281%2F3%29%283%29%29%2Ay=%28-3%2F1%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.

y=-3%2F3 Multiply the terms on the right side


y=-1 Reduce


So this is the other answer


y=-1<---------------------------------Other answer


So our solution is

x=5%2F2 and y=-1

which can also look like

(5%2F2,-1)

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

2%2Ax-5%2Ay=10
4%2Ax%2B3%2Ay=7

we get


graph of 2%2Ax-5%2Ay=10 (red) and 4%2Ax%2B3%2Ay=7 (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 (5%2F2,-1). This verifies our answer.


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Check:

Plug in (5%2F2,-1) into the system of equations


Let x=5%2F2 and y=-1. Now plug those values into the equation 2%2Ax-5%2Ay=10

2%2A%285%2F2%29-5%2A%28-1%29=10 Plug in x=5%2F2 and y=-1


10%2F2%2B5=10 Multiply


20%2F2=10 Add


10=10 Reduce. Since this equation is true the solution works.


So the solution (5%2F2,-1) satisfies 2%2Ax-5%2Ay=10



Let x=5%2F2 and y=-1. Now plug those values into the equation 4%2Ax%2B3%2Ay=7

4%2A%285%2F2%29%2B3%2A%28-1%29=7 Plug in x=5%2F2 and y=-1


20%2F2-3=7 Multiply


14%2F2=7 Add


7=7 Reduce. Since this equation is true the solution works.


So the solution (5%2F2,-1) satisfies 4%2Ax%2B3%2Ay=7


Since the solution (5%2F2,-1) satisfies the system of equations


2%2Ax-5%2Ay=10
4%2Ax%2B3%2Ay=7


this verifies our answer.