Step-by-step explanation:
Definition:
In systems of linear equations with two variables, the substitution method involves using the values of one equation into the other equation to solve for the value of one of the variables. Then, we substitute the value of that one variable into either one of the equations to solve for the value of the other variable. The goal is to solve for the solution to the given system.
Example:
I will demonstrate how the substitution method works. Suppose we have the following systems of linear equations with two variables:
Equation 1: 3x + 2y = 11
Equation 2: -x + y = 3
Staring with Equation 2, we must add x to both sides of the equation to isolate y:
Equation 2: -x + y = 3
-x + x + y = x + 3
y = x + 3
Next, substitute the value of y in Equation 2 (from the previous step), and substitute its value into Equation 1 to solve for x:
Equation 1: 3x + 2y = 11
Substitute y = x + 3 into Equation 1:
3x + 2(x + 3) = 11
Distribute 2 into the parenthesis:
3x + 2x + 6 = 11
Combine like terms:
5x + 6 = 11
Subtract 6 from both sides:
5x + 6 - 6 = 11 - 6
5x = 5
Divide both sides by 5 to solve for x:
[tex]\displaystyle\mathsf{\frac{5x}{5}=\:\frac{5}{5}}[/tex]
x = 1
Substitute the value of x into Equation 2 to solve for y:
Equation 2: -x + y = 3
-1 + y = 3
Add 1 to both sides to isolate y:
-1 + 1 + y = 3 + 1
y = 4
Therefore, the solution to the given systems of linear equations is x = 1, y = 4, or (1, 4).
Verify:
In order to verify whether we have the correct solution to the given system, substitute the values for x and y into both equations:
Equation 1: 3x + 2y = 11
3( 1 ) + 2(4) = 11
3 + 8 = 11
11 = 11 (True statement).
Equation 2: -x + y = 3
-( 1 ) + 4 = 3
-1 + 4 = 3
3 = 3 (True statement).
