SOLUTION: 6x+5y=150 9x+10y=270 Explain to me how to do this, using the Substitution method

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Question 698280: 6x+5y=150
9x+10y=270
Explain to me how to do this, using the Substitution method
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
6x%2B5y=150
9x%2B10y=270

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

6%2Ax%2B5%2Ay=150
9%2Ax%2B10%2Ay=270

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=150-6%2AxSubtract 6%2Ax from both sides

y=%28150-6%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=30-%286%2F5%29%2Ax Now we've fully isolated y

Since y equals 30-%286%2F5%29%2Ax we can substitute the expression 30-%286%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


9%2Ax%2B10%2Ahighlight%28%2830-%286%2F5%29%2Ax%29%29=270 Replace y with 30-%286%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

9%2Ax%2B10%2A%2830%29%2B10%28-6%2F5%29x=270 Distribute 10 to 30-%286%2F5%29%2Ax

9%2Ax%2B300-%2860%2F5%29%2Ax=270 Multiply



9%2Ax%2B300-12%2Ax=270 Reduce any fractions

9%2Ax-12%2Ax=270-300 Subtract 300 from both sides


9%2Ax-12%2Ax=-30 Combine the terms on the right side



-3%2Ax=-30 Now combine the terms on the left side.


cross%28%281%2F-3%29%28-3%2F1%29%29x=%28-30%2F1%29%281%2F-3%29 Multiply both sides by 1%2F-3. This will cancel out -3%2F1 and isolate x

So when we multiply -30%2F1 and 1%2F-3 (and simplify) we get



x=10 <---------------------------------One answer

Now that we know that x=10, lets substitute that in for x to solve for y

9%2810%29%2B10%2Ay=270 Plug in x=10 into the 2nd equation

90%2B10%2Ay=270 Multiply

10%2Ay=270-90Subtract 90 from both sides

10%2Ay=180 Combine the terms on the right side

cross%28%281%2F10%29%2810%29%29%2Ay=%28180%2F1%29%281%2F10%29 Multiply both sides by 1%2F10. This will cancel out 10 on the left side.

y=180%2F10 Multiply the terms on the right side


y=18 Reduce


So this is the other answer


y=18<---------------------------------Other answer


So our solution is

x=10 and y=18

which can also look like

(10,18)

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

6%2Ax%2B5%2Ay=150
9%2Ax%2B10%2Ay=270

we get


graph of 6%2Ax%2B5%2Ay=150 (red) and 9%2Ax%2B10%2Ay=270 (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 (10,18). This verifies our answer.


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Check:

Plug in (10,18) into the system of equations


Let x=10 and y=18. Now plug those values into the equation 6%2Ax%2B5%2Ay=150

6%2A%2810%29%2B5%2A%2818%29=150 Plug in x=10 and y=18


60%2B90=150 Multiply


150=150 Add


150=150 Reduce. Since this equation is true the solution works.


So the solution (10,18) satisfies 6%2Ax%2B5%2Ay=150



Let x=10 and y=18. Now plug those values into the equation 9%2Ax%2B10%2Ay=270

9%2A%2810%29%2B10%2A%2818%29=270 Plug in x=10 and y=18


90%2B180=270 Multiply


270=270 Add


270=270 Reduce. Since this equation is true the solution works.


So the solution (10,18) satisfies 9%2Ax%2B10%2Ay=270


Since the solution (10,18) satisfies the system of equations


6%2Ax%2B5%2Ay=150
9%2Ax%2B10%2Ay=270


this verifies our answer.