SOLUTION: Three times a first number minus the second number is 20. The sum of the two numbers is 48. Find the two numbers.

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Question 88236: Three times a first number minus the second number is 20. The sum of the two numbers is 48. Find the two numbers.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=first #, y=second #

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


Lets start with the given system of linear equations

3%2Ax-1%2Ay=20
1%2Ax%2B1%2Ay=48

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=20-3%2AxSubtract 3%2Ax from both sides

y=%2820-3%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-20%2B3%2Ax Now we've fully isolated y

Since y equals -20%2B3%2Ax we can substitute the expression -20%2B3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%28-20%2B3%2Ax%29%29=48 Replace y with -20%2B3%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%28-20%29%2B1%283%29x=48 Distribute 1 to -20%2B3%2Ax

1%2Ax-20%2B3%2Ax=48 Multiply



1%2Ax-20%2B3%2Ax=48 Reduce any fractions

1%2Ax%2B3%2Ax=48%2B20Add 20 to both sides


1%2Ax%2B3%2Ax=68 Combine the terms on the right side



4%2Ax=68 Now combine the terms on the left side.


cross%28%281%2F4%29%284%2F1%29%29x=%2868%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4%2F1 and isolate x

So when we multiply 68%2F1 and 1%2F4 (and simplify) we get



x=17 <---------------------------------One answer

Now that we know that x=17, lets substitute that in for x to solve for y

1%2817%29%2B1%2Ay=48 Plug in x=17 into the 2nd equation

17%2B1%2Ay=48 Multiply

1%2Ay=48-17Subtract 17 from both sides

1%2Ay=31 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%2831%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=31%2F1 Multiply the terms on the right side


y=31 Reduce


So this is the other answer


y=31<---------------------------------Other answer


So our solution is

x=17 and y=31

which can also look like

(17,31)

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

3%2Ax-1%2Ay=20
1%2Ax%2B1%2Ay=48

we get


graph of 3%2Ax-1%2Ay=20 (red) and 1%2Ax%2B1%2Ay=48 (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 (17,31). This verifies our answer.


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Check:

Plug in (17,31) into the system of equations


Let x=17 and y=31. Now plug those values into the equation 3%2Ax-1%2Ay=20

3%2A%2817%29-1%2A%2831%29=20 Plug in x=17 and y=31


51-31=20 Multiply


20=20 Add


20=20 Reduce. Since this equation is true the solution works.


So the solution (17,31) satisfies 3%2Ax-1%2Ay=20



Let x=17 and y=31. Now plug those values into the equation 1%2Ax%2B1%2Ay=48

1%2A%2817%29%2B1%2A%2831%29=48 Plug in x=17 and y=31


17%2B31=48 Multiply


48=48 Add


48=48 Reduce. Since this equation is true the solution works.


So the solution (17,31) satisfies 1%2Ax%2B1%2Ay=48


Since the solution (17,31) satisfies the system of equations


3%2Ax-1%2Ay=20
1%2Ax%2B1%2Ay=48


this verifies our answer.





So the two numbers are 17 and 31