SOLUTION: 3x-2y=7 5x+4y=8

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Question 107643This question is from textbook
: 3x-2y=7
5x+4y=8 This question is from textbook

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-2%2Ay=7
5%2Ax%2B4%2Ay=8

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

-2%2Ay=7-3%2AxSubtract 3%2Ax from both sides

y=%287-3%2Ax%29%2F-2 Divide both sides by -2.


Which breaks down and reduces to



y=-7%2F2%2B%283%2F2%29%2Ax Now we've fully isolated y

Since y equals -7%2F2%2B%283%2F2%29%2Ax we can substitute the expression -7%2F2%2B%283%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


5%2Ax%2B4%2Ahighlight%28%28-7%2F2%2B%283%2F2%29%2Ax%29%29=8 Replace y with -7%2F2%2B%283%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

5%2Ax%2B4%2A%28-7%2F2%29%2B4%283%2F2%29x=8 Distribute 4 to -7%2F2%2B%283%2F2%29%2Ax

5%2Ax-28%2F2%2B%2812%2F2%29%2Ax=8 Multiply



5%2Ax-14%2B6%2Ax=8 Reduce any fractions

5%2Ax%2B6%2Ax=8%2B14Add 14 to both sides


5%2Ax%2B6%2Ax=22 Combine the terms on the right side



11%2Ax=22 Now combine the terms on the left side.


cross%28%281%2F11%29%2811%2F1%29%29x=%2822%2F1%29%281%2F11%29 Multiply both sides by 1%2F11. This will cancel out 11%2F1 and isolate x

So when we multiply 22%2F1 and 1%2F11 (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%2B4%2Ay=8 Plug in x=2 into the 2nd equation

10%2B4%2Ay=8 Multiply

4%2Ay=8-10Subtract 10 from both sides

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

cross%28%281%2F4%29%284%29%29%2Ay=%28-2%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.

y=-2%2F4 Multiply the terms on the right side


y=-1%2F2 Reduce


So this is the other answer


y=-1%2F2<---------------------------------Other answer


So our solution is

x=2 and y=-1%2F2

which can also look like

(2,-1%2F2)

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

3%2Ax-2%2Ay=7
5%2Ax%2B4%2Ay=8

we get


graph of 3%2Ax-2%2Ay=7 (red) and 5%2Ax%2B4%2Ay=8 (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,-1%2F2). This verifies our answer.


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Check:

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


Let x=2 and y=-1%2F2. Now plug those values into the equation 3%2Ax-2%2Ay=7

3%2A%282%29-2%2A%28-1%2F2%29=7 Plug in x=2 and y=-1%2F2


6%2B2%2F2=7 Multiply


14%2F2=7 Add


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


So the solution (2,-1%2F2) satisfies 3%2Ax-2%2Ay=7



Let x=2 and y=-1%2F2. Now plug those values into the equation 5%2Ax%2B4%2Ay=8

5%2A%282%29%2B4%2A%28-1%2F2%29=8 Plug in x=2 and y=-1%2F2


10-4%2F2=8 Multiply


16%2F2=8 Add


8=8 Reduce. Since this equation is true the solution works.


So the solution (2,-1%2F2) satisfies 5%2Ax%2B4%2Ay=8


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


3%2Ax-2%2Ay=7
5%2Ax%2B4%2Ay=8


this verifies our answer.