SOLUTION: Solve each system of equations. check by substitution 4m + n = 17 3m - 4n = 27

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Question 120287: Solve each system of equations. check by substitution
4m + n = 17
3m - 4n = 27
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

4m+%2B+n+=+17
3m+-+4n+=+27
let m=x and n=y
solution:
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

4%2Ax%2B1%2Ay=17
3%2Ax-4%2Ay=27

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=17-4%2AxSubtract 4%2Ax from both sides

y=%2817-4%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=17-4%2Ax Now we've fully isolated y

Since y equals 17-4%2Ax we can substitute the expression 17-4%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-4%2Ahighlight%28%2817-4%2Ax%29%29=27 Replace y with 17-4%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-4%2A%2817%29-4%28-4%29x=27 Distribute -4 to 17-4%2Ax

3%2Ax-68%2B16%2Ax=27 Multiply



3%2Ax-68%2B16%2Ax=27 Reduce any fractions

3%2Ax%2B16%2Ax=27%2B68Add 68 to both sides


3%2Ax%2B16%2Ax=95 Combine the terms on the right side



19%2Ax=95 Now combine the terms on the left side.


cross%28%281%2F19%29%2819%2F1%29%29x=%2895%2F1%29%281%2F19%29 Multiply both sides by 1%2F19. This will cancel out 19%2F1 and isolate x

So when we multiply 95%2F1 and 1%2F19 (and simplify) we get



x=5 <---------------------------------One answer

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

3%285%29-4%2Ay=27 Plug in x=5 into the 2nd equation

15-4%2Ay=27 Multiply

-4%2Ay=27-15Subtract 15 from both sides

-4%2Ay=12 Combine the terms on the right side

cross%28%281%2F-4%29%28-4%29%29%2Ay=%2812%2F1%29%281%2F-4%29 Multiply both sides by 1%2F-4. This will cancel out -4 on the left side.

y=12%2F-4 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=5 and y=-3

which can also look like

(5,-3)

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

4%2Ax%2B1%2Ay=17
3%2Ax-4%2Ay=27

we get


graph of 4%2Ax%2B1%2Ay=17 (red) and 3%2Ax-4%2Ay=27 (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,-3). This verifies our answer.


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Check:

Plug in (5,-3) into the system of equations


Let x=5 and y=-3. Now plug those values into the equation 4%2Ax%2B1%2Ay=17

4%2A%285%29%2B1%2A%28-3%29=17 Plug in x=5 and y=-3


20-3=17 Multiply


17=17 Add


17=17 Reduce. Since this equation is true the solution works.


So the solution (5,-3) satisfies 4%2Ax%2B1%2Ay=17



Let x=5 and y=-3. Now plug those values into the equation 3%2Ax-4%2Ay=27

3%2A%285%29-4%2A%28-3%29=27 Plug in x=5 and y=-3


15%2B12=27 Multiply


27=27 Add


27=27 Reduce. Since this equation is true the solution works.


So the solution (5,-3) satisfies 3%2Ax-4%2Ay=27


Since the solution (5,-3) satisfies the system of equations


4%2Ax%2B1%2Ay=17
3%2Ax-4%2Ay=27


this verifies our answer.