Solving systems of linear equations using the substitution method involves manipulating one of the equations to isolate one of the variables. Then, substituting that equation into the other equation(s) to eliminate the chosen variable. This ultimately results in having a single equation with only one variable, which can then easily be solved. Once that variable is found, substitute its value back into the equation you've isolated initially to find the other variable.
Let's work with a very simple example:
Example 1:
Given the system of linear equations:
1. x + y = 5
2. x - y = 1
Step 1: Isolate one of the variables
From equation 1, we can isolate the x variable as follows:
x = 5 - y
Step 2: Substitute the isolated equation into the other equation
Now, replace the x in equation 2 with the expression we got from step 1:
(5 - y) - y = 1
Step 3: Solve the equation resulting from the substitution
4 - 2y = 1
-2y = -3
y = 3/2
Step 4: Substitute the value of y back into the initial equation with the isolated variable
x = 5 - (3/2)
x = 5 - 1.5
x = 3.5
The solution of the system of linear equations is x = 3.5 and y = 1.5.
Keep in mind there are three possible outcomes when solving a system of linear equations using substitution:
1. One unique solution (as shown in the example above)
2. No solution (when the equations represent parallel lines)
3. Infinitely many solutions (when the equations represent the same line)
Solving systems using substitution
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