Solving systems using substitution
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)
To solve a system of equations using the substitution method, follow these steps:
1. Start by solving one of the equations for one variable in terms of the other variable. Choose the equation that seems easier to isolate a variable.
2. Substitute the expression obtained in step 1 for the equivalent variable in the other equation. This will create a new equation with only one variable.
3. Solve the new equation from step 2 for the remaining variable.
4. Substitute the value obtained in step 3 back into the original equation to find the value of the other variable.
5. Write the solution as an ordered pair (x, y), where x is the value of one variable and y is the value of the other variable.
Let's work through an example to demonstrate the process:
Example: Solve the system of equations
Equation 1: 2x + y = 9
Equation 2: x - y = 1
Step 1:
Let's solve Equation 2 for x:
x = y + 1
Step 2:
Substitute the expression for x into Equation 1:
2(y + 1) + y = 9
Simplify: 2y + 2 + y = 9
Combine like terms: 3y + 2 = 9
Step 3:
Solve the new equation for y:
Subtract 2 from both sides: 3y = 7
Divide both sides by 3: y = 7/3
Step 4:
Substitute y = 7/3 into the expression for x:
x = (7/3) + 1
x = 10/3
Step 5:
Write the solution as an ordered pair:
(x, y) = (10/3, 7/3)
So, the solution to the system of equations is (10/3, 7/3).
Solving systems of equations using the substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Here are the steps to solve a system of equations using substitution:
1. Identify the system of equations. For example, let's consider the following system:
Equation 1: 2x + y = 7
Equation 2: 3x - 2y = 1
2. Solve one of the equations for one variable. Pick one of the equations and solve it explicitly for one of the variables. Let's solve Equation 1 for y:
From Equation 1: y = 7 - 2x
3. Substitute the expression found in step 2 into the other equation. Take the expression you found in step 2 and substitute it into the other equation. In this case, substitute y = 7 - 2x into Equation 2:
Substituting y in Equation 2: 3x - 2(7 - 2x) = 1
4. Simplify and solve the equation obtained in step 3. Simplify the equation obtained in step 3 and solve for the remaining variable. In this example, let's solve for x:
Expanding Equation 2: 3x - 14 + 4x = 1
Combining like terms: 7x - 14 = 1
Add 14 to both sides: 7x = 15
Divide by 7: x = 15/7
5. Substitute the value found in step 4 back into the expression from step 2. Once you have found the value for one variable, substitute it back into one of the original equations to solve for the remaining variable. Let's substitute x = 15/7 into Equation 1:
Substituting x in Equation 1: 2(15/7) + y = 7
Simplifying: 30/7 + y = 7
Subtract 30/7 from both sides: y = 7 - 30/7
6. Simplify and find the value of the remaining variable. Simplify the equation obtained in step 5 and solve for the remaining variable. In this example, let's solve for y:
Common denominator: y = (49 - 30) / 7 = 19/7
So the solution to the system of equations is x = 15/7, y = 19/7.