# X and Y Substitution - Easier, Fewer Checks

### Provide the value for the missing coordinate.

* Locked after checks
All Correct Answers shown after locked.

First of all.  You are MUCH smarter than this program.
What you would normally write as:
y - 3x = 2
The equation maker might write as
-1y + -3x = 2
or occasionally as
-1y + -3x + 0 = 2
That's okay.  We know you're smart enough to ignore the superfluous information.

Solving by substitution
Any equation with two variables can be solved by coming up with a value for one of the variables, substituting it in, and then solving for the variable that remains.

Example 1:

2x + -3 = -1y
If x is 2, then
2(2) + -3 = -1y    [Substitution]
4 + -3 = -1y        [Simplification]
-1=-1y                [Simplification]
1 = y                    [Division Property of Equality (divided both sides by -1)]

For the same equation, if y is 1

2x + -3 = -1(1)    [Substitution]
2x + -3 = -1        [Simplification]
2x = 2                [Addition Property of Equality (added 3 to both sides)]
x = 1                    [Division Prop. of Equality (divided both sides by 2]

Example 2:

-2x = 3y + 6
If x is -3, then
-2(-3) = 3y + 6    [Substitution]
6 = 3y + 6       [Simplification]
0 = 3y                [Addition Property of Equality (added -6 to both sides)]
0 = y                    [Division Property of Equality (divided both sides by 3)]

If y is -2, then
-2x = 3(-2) + 6    [Substitution]
-2x = -6 + 6       [Simplification]
-2x = 0                [Simplification]
x = 0                    [Division Property of Equality (divided both sides by -2)]

NOTE:
If x is 4 then
-2(4) = 3y + 6    [Substitution]
-8 = 3y + 6       [Simplification]
-14 = 3y                [Addition Property of Equality (added -6 to both sides)]
-14/3 =  y                    [Division Property of Equality (divided both sides by 3)]

THE PROCESS IS THE SAME, but it is possible to get fractional answers
depending on what values you use for your x and y.

You should also note that any pair of answers (coordinates) that come from a single linear equation are represented by a line in the coordinate plane.  That is every pair that works in the equation will land on the same line.  It also means that once you know what the line looks like that every pair of points that you see on the line will also work in the equation.