Question 844785: I have three numbers. the sum of the first two minus the third is -1. The sum of the first and third is minus the second is 13. The sum of all three numbers is 17. What are my numbers?
Answer by pmesler(52)   (Show Source):
You can put this solution on YOUR website! First, let's look at what the question is asking. It's asking about three unknown numbers. We know that unknowns can be expressed as variables. We can express these variables as x, y, and z.
Next, we need to turn the first sentence into a mathematical statement. Let's look at the first statement: "the sum of the first two minus the third is -1."
If we write this mathematically it would look like this:
x+y-z = -1.
The second statement says "The sum of the first and third is minus the second is 13," and we can write it thus:
x-y+z = 13.
The third statement says "The sum of all three numbers is 17." We can write that like this:
x+y+z=17
What we now have is a system of equations in three variables. Generally, in mathematics, the number of variables will dictate the number of equations necessary to solve for the unknowns. In this case, since there are 3 unknowns, x, y, z, we need three equations to solve for these values.
Let's write them like this:
x+y-z = -1
x-y+z = 13
x+y+z = 17
There are several ways to approach this problem, but one of the easiest methods is called the substitution method.
Essentially we need to simplify the expressions so we can isolate each variable so we can get their individual values.
To begin let's add the first two equations to give us our second equation. When we do this we get the following new system of equations:
x+y-z = -1
2x = 12 <--------> (x+x)+ (y-y) + (-1+13)
x+y+z = 17
Since 2x = 12, right away we can see that x = 6. Now all we have to do is solve for y and z. To do that we need to add the first equation by the third to obtain a new third equation. When we do that we get this:
x+y-z = -1
x = 6
2x+2y = 16 <---------> (x+x) + (y+y) (-1+17)
Now, let's simplify the third equation by dividing each side by 2. We then obtain the following:
x+y = 8.
Now, we know that x = 6 from the previous equation. So, we simply substitute the value of x into this equation and solve for y.
6+y = 8. Therefore y = 2.
Now to find the value for z we substitute the value for x and y into the ORIGINAL third equation x+y+z = 17, and we get
6+2+z = 17
8 + z = 17. Therefore z = 9.
The solution is (6,2,9). To verify that this correct, let's substitute the values into all three equations and see if they check out.
Equation 1 <--------> x+y-z = -1 <--------> 6+2-9 = -1 = 8-9 = -1. The first equation is correct. Now for the second.
Equation 2 <-------> x-y+z = 13 <--------> 6-2+9 = 13 = 4+9 = 13. The second equation is correct. Now for the third.
Equation 3 <-------> x+y+z = 17 <----------> 6+2+9 =17. The third equation is correct.
The problem is solved.
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