Solving Systems of Equations by Elimination

Welcome back to Professor Baker's Math Class! Today, November 18th, 2013, we delved into the elimination method for solving systems of linear equations. This method provides a systematic way to find the values of variables that satisfy multiple equations simultaneously.

Key Concepts of Elimination

The elimination method relies on manipulating equations to eliminate one variable, allowing us to solve for the remaining variable. Here's a breakdown of the process:

  1. Align: Ensure the equations are aligned, with like terms (x terms, y terms, constants) stacked vertically.
  2. Multiply (if necessary): Multiply one or both equations by a constant so that the coefficients of one variable are opposites (e.g., 2x and -2x). This is a crucial step for setting up the elimination.
  3. Add: Add the equations together. The chosen variable should now be eliminated, leaving you with a single equation in one variable.
  4. Solve: Solve the resulting equation for the remaining variable.
  5. Substitute: Substitute the value you found back into either of the original equations to solve for the other variable.
  6. Check: Verify your solution by substituting both values into both original equations. This ensures the solution satisfies the entire system.

Example Problems

Let's look at an example system and solve it using elimination:

$$\begin{cases} -3x + 2y = -6 \\ 5x - 2y = 18 \end{cases}$$

Notice that the 'y' terms have opposite coefficients (+2 and -2). We can proceed directly to adding the equations:

$(-3x + 2y) + (5x - 2y) = -6 + 18$$ This simplifies to:

$2x = 12$

Dividing both sides by 2, we get:

$x = 6$

Now, substitute x = 6 into either of the original equations. Let's use the first equation:

$-3(6) + 2y = -6$

$-18 + 2y = -6$

$2y = 12$

$y = 6$

Therefore, the solution to the system is $(6, 6)$. We should always check our work! Substituting into both original equations:

-3(6) + 2(6) = -18 + 12 = -6 (Correct!)

5(6) - 2(6) = 30 - 12 = 18 (Correct!)

As you can see in other examples from class notes, sometimes you need to multiply one or both equations before adding. Consider this example:

$$\begin{cases} 2x - 5y = 10 \\ -3x + 4y = -15 \end{cases}$$

To eliminate 'x', we can multiply the first equation by 3 and the second equation by 2:

$$\begin{cases} 6x - 15y = 30 \\ -6x + 8y = -30 \end{cases}$$

Adding the equations gives us: -7y = 0, so y = 0.

Substituting y = 0 into the first original equation: 2x - 5(0) = 10, so 2x = 10 and x = 5.

The solution is (5, 0). Check your work!

Homework

For homework, please complete the entire worksheet provided in class. Practice is key to mastering the elimination method. Remember to show all your steps clearly and double-check your answers!

Keep up the great work, and see you in the next class!