1.6 Solving Systems of Equations by Elimination Method
The method of elimination is a technique used to solve systems of linear equations. This method involves adding or subtracting the equations to eliminate one of the variables. Here’s a step-by-step process:
Step 1: Write Down the System of Equations
Write down the system of equations that you want to solve. For example, let’s consider the following system of equations:
\(2x + 3y = 12\)
\(4x - 3y = 6\)
Step 2: Arrange the Equations
Arrange the equations such that like terms are vertically aligned. In our example, the equations are already arranged properly.
Step 3: Eliminate One Variable
Add or subtract the equations to eliminate one of the variables. In our example, if we add equation (1) and equation (2), we get:
\((2x + 3y) + (4x - 3y) = 12 + 6\)
Solving this gives:
\(6x = 18\)
Step 4: Solve for the Eliminated Variable
Solve the resulting equation for the eliminated variable. In our example, solving for '\(x\)' gives:
\(x = 3\)
Step 5: Substitute the Value into One of the Original Equations
Substitute the value of '\(x\)' into one of the original equations. Substituting \(x = 3\) into equation (1) gives:
\(2(3) + 3y = 12\)
Step 6: Solve for the Other Variable
Solve the equation for the other variable. Solving for '\(y\)' gives:
\(y = 2\)
Step 7: Check Your Solution
Substitute the values of '\(x\)' and '\(y\)' into both original equations to check if they are correct. In our example, substituting \(x = 3\) and \(y = 2\) into both equations verifies that they are the correct solutions.
So, the solution to the system of equations is \(x = 3\) and \(y = 2\).
Example 2: Consider the following system of equations:
\(3x + 2y = 10\)
\(5x - 4y = 2\)
Step 1: Make the Coefficients of One Variable Equal
To use the elimination method, we need the coefficients of one of the variables to be equal in both equations. We can achieve this by multiplying both sides of equation (1) by 2:
\(6x + 4y = 20\)
\(5x - 4y = 2\)
Step 2: Add or Subtract the Equations
Now, add equation (3) and equation (2) to eliminate:
\((6x + 4y) + (5x - 4y) = 20 + 2\)
Solving this gives:
\(11x = 22\)
Step 3: Solve for the Eliminated Variable
Solving for '\(x\)' gives:
\(x = 2\)
Step 4: Substitute the Value into One of the Original Equations
Substitute \(x = 2\) into equation (1):
\(3(2) + 2y = 10\)
\(6 + 2y = 10\)
\(2y = 10 - 6\)
\(2y = 4\)
Step 5: Solve for the Other Variable
Solving for '\(y\)' gives:
\(y = 2\)
Step 6: Check Your Solution
Substitute \(x = 2\) and \(y = 2\) into both original equations to check if they are correct.
So, the solution to the system of equations is \(x = 2\) and \(y = 2\).