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Solve the following system of equations by the substitution method.x - y = 0x - y - 2 = 0What is the solution set?

Solve the following system of equations by the substitution method.

x - y = 0

x - y - 2 = 0

What is the solution set?

Hi Kayla,

Kevin and John are right that there are no solutions to this system. But using substitution to see that is a little weird here, even though the problem asked you to use substitution. Here's a faster way to see the system has no solutions:

The first equation just tells you x-y=0. But the second equation just tells you x-y=2, if you add 2 to both sides. So you're looking for points (x,y) that satisfy both x-y=0 and x-y=2. That's impossible! If x-y=0, then x-y can't equal 2. So there's no way to find points that satisfy both equations.

Another way to see this system has no solutions is to think graphically. If you put each of the equations in slope-intercept form, you get y=x for the first equation and y=x-2 for the second equation. But these lines have the same slope (namely, a slope of 1)! That means if you graph them, they will be parallel --- so they will never intersect! Because the lines don't intersect, there are no points that lie on both of them. And that's another way of saying there is no point (x,y) that satisfies both equations.

Even if your teacher asks you to think about a problem in a certain way (as in this problem with the substitution method), you should always try to think about a problem in many different ways -- each perspective provides a different kind of insight. And the more insight you have into a problem from many different perspectives, the more you really understand it.

The first step in solving a system of equations by substitution is solving one of the equations for one of the variables

x - y = 0                Equation 1

x - y + y = 0 +y      Add y to each side

x = y                     Simplify

After solving an equation for one of the variables, substitute that value into the other equation.  In this problem, I can substitute y for x.

x - y - 2 = 0                  Equation 2

y - y - 2 = 0                  Substitute the value of x from equation 1

0 - 2 = 0                      Inverse Property of addition

-2 = 0                          Identity Property of addition

Since -2 can not equal 0, there is no solution.  The lines are parallel.  It is an inconsistent system.

Since x - y = 0, we can solve for x, to get
x = y.

We now substitute that into the second equation to get
y - y - 2 = 0 or
-2 = 0

which is a false statement and is never true.

Therefore, there is no solution set for this system.