# SOLUTION: The sum of two numbers is 50 and their difference is 18. Find the numbers. Im so confused could someone please help?

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Question 120563: The sum of two numbers is 50 and their difference is 18. Find the numbers.

Found 2 solutions by jim_thompson5910, solver91311:
You can put this solution on YOUR website!
If the "sum of two numbers is 50", then we have the first equation

If "their difference is 18", then we have the second equation

 Solved by pluggable solver: Solving a linear system of equations by subsitution Lets start with the given system of linear equations 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 Subtract from both sides Divide both sides by 1. Which breaks down and reduces to Now we've fully isolated y Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x. Replace y with . Since this eliminates y, we can now solve for x. Distribute -1 to Multiply Reduce any fractions Add to both sides Combine the terms on the right side Now combine the terms on the left side. Multiply both sides by . This will cancel out and isolate x So when we multiply and (and simplify) we get <---------------------------------One answer Now that we know that , lets substitute that in for x to solve for y Plug in into the 2nd equation Multiply Subtract from both sides Combine the terms on the right side Multiply both sides by . This will cancel out -1 on the left side. Multiply the terms on the right side Reduce So this is the other answer <---------------------------------Other answer So our solution is and which can also look like (,) Notice if we graph the equations (if you need help with graphing, check out this solver) we get graph of (red) and (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 (,). This verifies our answer. ----------------------------------------------------------------------------------------------- Check: Plug in (,) into the system of equations Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Since the solution (,) satisfies the system of equations this verifies our answer.

So the two numbers are 34 and 16

You can put this solution on YOUR website!
Let's call one of the numbers x and the other one y. Their sum is then , and their difference is .

The sum of two numbers, , is, (=), 50, so .

The difference of the same two numbers, , is, (=), 18, so

Now you have a system of two equations in two variables. There are a number of methods to solve these systems, but this one lends itself to the Gaussian Elimination method. In this case, just add the two equations term by term which will eliminate the y variable, thus:

So one of the numbers has to be 34. You can get the other just by subtracting 34 from 50 because of the relationship , but I'm going to show the elimination method on the original two equations to eliminate the x variable this time.

Step 1: Multiply the bottom equation by -1

Now add them term by term: