What does it mean to solve by subsitution and elimination

It gave these 2 problems:

Solve by substitution:
4x - 3y = 5
y = -3x + 7


Solve by elimination:
x - 2y = 6
3x - y = -2



What exactly does it mean by "substitution" and "elimination"?
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To solve by substitution and elimination means to solve for x and y:
4x - 3y = 5
y = -3x + 7

Solve by substitution (this means we substitute -3x + 7 for y) :
4x - 3(-3x + 7) = 5
4x + 9x - 21 = 5
13x = 26
13x/13 = 26/13
x = 2

Now that we know that x = 2 we substitute it in the 4x - 3y = 5 equation so we can solve for y:

4(2) - 3y = 5
8 - 3y = 5
-3y = -3
-3y/-3 = -3/-3
y = 1

So for the first equation x = 2 and y = 1.






To Solve by elimination with the second equation you need to isolate one variable by means of multiplication and subtraction:

x - 2y = 6
3x - y = -2

3(x - 2y = 6) = 3x - 6y = 18
-(3x - y = -2) = -3x + y = +2


3x - 6y = 18
-3x + y = +2
-5y = 20
-5y/-5 = 20/-5
y = -4

x - 2y = 6
3x - y = -2


x - 2y = 6
-2(3x - y = -2) = -6x + 2y = 4

x - 2y = 6
-6x + 2y = 4
-5x = 10
-5x/-5 =10/-5
x = -2

So here x = -2 and y = -4.

Substitution:
Solve one equation for x or y. Use that result to replace that variable in the other equation. This leaves a single equation in one variable.

In this problem, the second equation is already solved for y. Replace y in the first equation with (-3x + 7).
4x - 3(-3x + 7) = 5
x = 2
y = 1


Elimination:
Multiply one of the equations by a constant so that one of its coefficients matches the other equation. Then subtract the two equations. This eliminates one variable.