SOLUTION: Use substitution to solve: 2x+y=7,5x-2y=4

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Question 111724: Use substitution to solve: 2x+y=7,5x-2y=4
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%2B1%2Ay=7
5%2Ax-2%2Ay=4

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

1%2Ay=7-2%2AxSubtract 2%2Ax from both sides

y=%287-2%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=7-2%2Ax Now we've fully isolated y

Since y equals 7-2%2Ax we can substitute the expression 7-2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


5%2Ax%2B-2%2Ahighlight%28%287-2%2Ax%29%29=4 Replace y with 7-2%2Ax. Since this eliminates y, we can now solve for x.

5%2Ax-2%2A%287%29-2%28-2%29x=4 Distribute -2 to 7-2%2Ax

5%2Ax-14%2B4%2Ax=4 Multiply



5%2Ax-14%2B4%2Ax=4 Reduce any fractions

5%2Ax%2B4%2Ax=4%2B14Add 14 to both sides


5%2Ax%2B4%2Ax=18 Combine the terms on the right side



9%2Ax=18 Now combine the terms on the left side.


cross%28%281%2F9%29%289%2F1%29%29x=%2818%2F1%29%281%2F9%29 Multiply both sides by 1%2F9. This will cancel out 9%2F1 and isolate x

So when we multiply 18%2F1 and 1%2F9 (and simplify) we get



x=2 <---------------------------------One answer

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

5%282%29-2%2Ay=4 Plug in x=2 into the 2nd equation

10-2%2Ay=4 Multiply

-2%2Ay=4-10Subtract 10 from both sides

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

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

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


y=3 Reduce


So this is the other answer


y=3<---------------------------------Other answer


So our solution is

x=2 and y=3

which can also look like

(2,3)

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

2%2Ax%2B1%2Ay=7
5%2Ax-2%2Ay=4

we get


graph of 2%2Ax%2B1%2Ay=7 (red) and 5%2Ax-2%2Ay=4 (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 (2,3). This verifies our answer.


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Check:

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


Let x=2 and y=3. Now plug those values into the equation 2%2Ax%2B1%2Ay=7

2%2A%282%29%2B1%2A%283%29=7 Plug in x=2 and y=3


4%2B3=7 Multiply


7=7 Add


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


So the solution (2,3) satisfies 2%2Ax%2B1%2Ay=7



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

5%2A%282%29-2%2A%283%29=4 Plug in x=2 and y=3


10-6=4 Multiply


4=4 Add


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


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


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


2%2Ax%2B1%2Ay=7
5%2Ax-2%2Ay=4


this verifies our answer.