What does it mean to solve by subsitution and elimination

The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other.

The method of solving "by elimination" works by 'eliminating' one variable from the system of equations and then substituting this value back into one of the equations and solving for the remaining variable left over.

1.) 4x-3y = 5
y = -3x+7

4x-3(-3x+7) = 5
4x+9x-27 = 5
13x-27 = 5
13x = 32
x = 32/13

y = -3x+7
y = -3(32/13)+7
y = -96/13+7
y = -5/13
Solutions: x = 32/13, y = -5/13; (32/13,-5/13)

2.)
x-2y = 6
3x-y = -2
+_______

x+2y = 6
2{3x-y = -2}
+_________

x+2y = 6
6x-2y = -2
+________
7x+0y = 4

7x = 4
x = 4/7

x+2y = 6
(4/7)+2y = 6
4/7+2y = 6
2y = 38/7
y = 19/7
Solutions: x = 4/7, y = 19/7; (4/7,19/7)

solve for one variable in terms of the other variable, then sub that into the second equation and solve for one variable, them plug that value into either equation to solve for the second.

4x-3(-3x+7)=5
4x+9x-21=5
13x=26
x=2(plug this into either equation and solve for y)
y=-3(2)+7
y=1