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# How do you write a system of linear equations with two variables?

what is it and how do you write the problem

Hi Carolyn,

A great question. A system of equations ("system" meaning more than one equation) with two variables looks like this:

x + y= 8

2x + 3y = 5

There are several methods to solving systems of equations, including "substitution," "elimination," or graphing. The easiest ways involve substitution or elimination. In the set above, notice that the first equation "x + y = 8" does not have any coefficients in front of the "x" or the "y." When comparing this to the equation below, "2x + 3y = 5," it is important to decide which term to eliminate, either the one containing "x" or the one containing "y." For our purposes, let's choose to eliminate the "x" in both equations. For this, you will need to multiply the first equation by [-2]. Why? Multiplying by [-2] will make the top term [-2x], which, when we add to the bottom term, [2x], will cause it to cancel or "zero" out. So, Let's try it:

-2 [ x + y = 8 ] +

2x + 3y = 5

==>>    Equation 1: -2x - 2y = -16

Equation 2:   2x + 3y = 5

======>> Add both equations to yield one final equation: [ y = -11 ]

Great! Now you know what one of the variables in the system of equations is! Use this to substitute the variable on either one of the original equations: (I like the top one best, less work!)

x + y = 8, where y = -11.

x + (-11) = 8 <=== add (11) to both sides of the equal sign:

x = 19.

The final answer? x = 19, y = -11

You're done!

I hope this helps you!

You aren't done until you check your work, to make sure you haven't made any arithmetic errors.

(x, y) = (19, -11)

x + y    = 8                                2x + 3y       = 5

19 - 11 = 8                           2(19) + 3(-11) = 5

8   = 8                               38   -   33    = 5

check                                               5        = 5

check

Now, assuming you haven't made an arithmetic mistake in checking that matches a mistake in the solution (unlikely), you know the answer is correct.