SOLUTION: I need help solving this system by substitution and elimination. -16x+45y=134 40x+18y=-45

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Question 112352: I need help solving this system by substitution and elimination.
-16x+45y=134
40x+18y=-45

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

-16%2Ax%2B45%2Ay=134
40%2Ax%2B18%2Ay=-45

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

45%2Ay=134%2B16%2AxAdd 16%2Ax to both sides

y=%28134%2B16%2Ax%29%2F45 Divide both sides by 45.


Which breaks down and reduces to



y=134%2F45%2B%2816%2F45%29%2Ax Now we've fully isolated y

Since y equals 134%2F45%2B%2816%2F45%29%2Ax we can substitute the expression 134%2F45%2B%2816%2F45%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


40%2Ax%2B18%2Ahighlight%28%28134%2F45%2B%2816%2F45%29%2Ax%29%29=-45 Replace y with 134%2F45%2B%2816%2F45%29%2Ax. Since this eliminates y, we can now solve for x.

40%2Ax%2B18%2A%28134%2F45%29%2B18%2816%2F45%29x=-45 Distribute 18 to 134%2F45%2B%2816%2F45%29%2Ax

40%2Ax%2B2412%2F45%2B%28288%2F45%29%2Ax=-45 Multiply



40%2Ax%2B268%2F5%2B%2832%2F5%29%2Ax=-45 Reduce any fractions

40%2Ax%2B%2832%2F5%29%2Ax=-45-268%2F5 Subtract 268%2F5 from both sides


40%2Ax%2B%2832%2F5%29%2Ax=-225%2F5-268%2F5 Make -45 into a fraction with a denominator of 5


40%2Ax%2B%2832%2F5%29%2Ax=-493%2F5 Combine the terms on the right side



%28200%2F5%29%2Ax%2B%2832%2F5%29x=-493%2F5 Make 40 into a fraction with a denominator of 5

%28232%2F5%29%2Ax=-493%2F5 Now combine the terms on the left side.


cross%28%285%2F232%29%28232%2F5%29%29x=%28-493%2F5%29%285%2F232%29 Multiply both sides by 5%2F232. This will cancel out 232%2F5 and isolate x

So when we multiply -493%2F5 and 5%2F232 (and simplify) we get



x=-17%2F8 <---------------------------------One answer

Now that we know that x=-17%2F8, lets substitute that in for x to solve for y

40%28-17%2F8%29%2B18%2Ay=-45 Plug in x=-17%2F8 into the 2nd equation

-85%2B18%2Ay=-45 Multiply

18%2Ay=-45%2B85Add 85 to both sides

18%2Ay=40 Combine the terms on the right side

cross%28%281%2F18%29%2818%29%29%2Ay=%2840%2F1%29%281%2F18%29 Multiply both sides by 1%2F18. This will cancel out 18 on the left side.

y=40%2F18 Multiply the terms on the right side


y=20%2F9 Reduce


So this is the other answer


y=20%2F9<---------------------------------Other answer


So our solution is

x=-17%2F8 and y=20%2F9

which can also look like

(-17%2F8,20%2F9)

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

-16%2Ax%2B45%2Ay=134
40%2Ax%2B18%2Ay=-45

we get


graph of -16%2Ax%2B45%2Ay=134 (red) and 40%2Ax%2B18%2Ay=-45 (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 (-17%2F8,20%2F9). This verifies our answer.


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Check:

Plug in (-17%2F8,20%2F9) into the system of equations


Let x=-17%2F8 and y=20%2F9. Now plug those values into the equation -16%2Ax%2B45%2Ay=134

-16%2A%28-17%2F8%29%2B45%2A%2820%2F9%29=134 Plug in x=-17%2F8 and y=20%2F9


272%2F8%2B900%2F9=134 Multiply


9648%2F72=134 Add


134=134 Reduce. Since this equation is true the solution works.


So the solution (-17%2F8,20%2F9) satisfies -16%2Ax%2B45%2Ay=134



Let x=-17%2F8 and y=20%2F9. Now plug those values into the equation 40%2Ax%2B18%2Ay=-45

40%2A%28-17%2F8%29%2B18%2A%2820%2F9%29=-45 Plug in x=-17%2F8 and y=20%2F9


-680%2F8%2B360%2F9=-45 Multiply


-3240%2F72=-45 Add


-45=-45 Reduce. Since this equation is true the solution works.


So the solution (-17%2F8,20%2F9) satisfies 40%2Ax%2B18%2Ay=-45


Since the solution (-17%2F8,20%2F9) satisfies the system of equations


-16%2Ax%2B45%2Ay=134
40%2Ax%2B18%2Ay=-45


this verifies our answer.