Solving Linear Systems with Fractions: Elimination Method

Linear systems are a fundamental concept in algebra, and they involve solving for multiple unknown variables in a set of equations. When these equations contain fractions, solving the system can seem more complex. However, the elimination method provides a systematic way to tackle such problems.

Understanding the Elimination Method

The elimination method, as the name suggests, aims to eliminate one variable from the system of equations. This is achieved by manipulating the equations to create opposite coefficients for one of the variables. By adding the equations together, the variable with opposite coefficients cancels out, leaving you with a single equation in one variable. This equation can then be solved to find the value of that variable.

Steps to Solve Linear Systems with Fractions

Let's break down the process of solving linear systems with fractions using the elimination method:

1. Clear the Fractions: Multiply each equation by the least common multiple (LCM) of the denominators of the fractions in that equation. This will eliminate the fractions and make the equations easier to work with.
2. Choose a Variable to Eliminate: Select one of the variables you want to eliminate. Examine the coefficients of that variable in both equations. If they are not opposites, multiply one or both equations by appropriate constants to create opposite coefficients for the chosen variable.
3. Add the Equations: Add the two modified equations together. This step should eliminate the variable you chose in step 2, leaving you with an equation in one variable.
4. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
5. Substitute to Find the Other Variable: Substitute the value you just found back into either of the original equations (before clearing the fractions). Solve this equation for the other variable.
6. Check Your Solution: Substitute the values you found for both variables back into the original equations to ensure they satisfy both equations. This step verifies that your solution is correct.

Example

Let's consider the following system of equations:

Equation 1: (1/2)x + (2/3)y = 1

Equation 2: (3/4)x - (1/2)y = 2

Step 1: Clear the Fractions:

• Multiply Equation 1 by 6 (LCM of 2 and 3): 3x + 4y = 6
• Multiply Equation 2 by 4 (LCM of 4 and 2): 3x - 2y = 8

Step 2: Choose a Variable to Eliminate:

Let's eliminate 'x'. The coefficients of 'x' are already the same (3). To make them opposites, we can multiply Equation 2 by -1.

Step 3: Add the Equations:

• 3x + 4y = 6
• -3x + 2y = -8
• -----------------
• 6y = -2

Step 4: Solve for the Remaining Variable:

Divide both sides by 6: y = -1/3

Step 5: Substitute to Find the Other Variable:

Substitute y = -1/3 into either of the original equations. Let's use Equation 1:

(1/2)x + (2/3)(-1/3) = 1

Simplify and solve for x:

(1/2)x - 2/9 = 1

(1/2)x = 11/9

x = 22/9

Step 6: Check Your Solution:

Substitute x = 22/9 and y = -1/3 into both original equations to verify that they satisfy both equations.

Conclusion

The elimination method is a powerful tool for solving linear systems, even when fractions are involved. By following these steps, you can systematically eliminate variables and solve for the unknown values. This method is essential for various applications in mathematics, science, and engineering.