Understanding and Solving Simultaneous Equations Using the Elimination Method

When faced with a pair of linear equations, such as the one presented below, the elimination method is a powerful and straightforward technique to determine the values of variables involved. Letrsquo;s explore the process in detail.

Introduction to Simultaneous Equations

Simultaneous equations are a set of equations containing multiple variables that must be solved simultaneously. The equations presented here are:

2x - 3y 11 5x - 2y 18

Our goal is to find the values of x and y that satisfy both equations at the same time.

The Elimination Method

The elimination method works by eliminating one variable so that we are left with a single equation in one variable. Here's how we can do it step-by-step:

Step 1: Align the Equations

Write the equations side by side:

2x - 3y 11 5x - 2y 18

Step 2: Eliminate One Variable

To eliminate y, we need to make the coefficients of y in both equations opposites. We can achieve this by multiplying the first equation by 2 and the second equation by 3:

2(2x - 3y) 2(11) → 4x - 6y 22

3(5x - 2y) 3(18) → 15x - 6y 54

Step 3: Add the Equations

Now, add the modified equations to eliminate y:

4x - 6y 15x - 6y 22 54

19x 76

Step 4: Solve for x

Divide both sides by 19 to find the value of x:

x frac{76}{19} 4

Step 5: Substitute x Back to Find y

Now, substitute x 4 back into one of the original equations, such as the first one:

2(4) - 3y 11

8 - 3y 11

-3y 11 - 8

-3y 3

y -1

Final Solution

The solution to the simultaneous equations is:

x 4 and y -1

This means the ordered pair (x, y) is (4, -1).

Additional Tips for Solving Simultaneous Equations

Choose the Proper Method: While the elimination method is effective in this case, other methods such as substitution can also be used, depending on the equations involved. Check Your Answer: Always substitute the values of x and y back into the original equations to ensure they satisfy both. Practice Regularly: The more you practice, the more comfortable you will become with these types of problems.

By mastering the elimination method and understanding how to apply it to various simultaneous equations, you can solve complex algebraic problems with confidence and efficiency.