Welcome back to Professor Baker's Math Class! Today, we are tackling a powerful technique for solving systems of linear equations: The Elimination Method (sometimes called Linear Combination).

While the Substitution Method is great for specific scenarios, Elimination is often the fastest way to solve standard systems, especially when the equations are already lined up nicely. The goal is simple: manipulate the equations so that when you add them together, one variable disappears (is eliminated), allowing you to solve for the other.

The 3 Steps to Success

According to our class notes, here is the roadmap to solving these problems:

  1. Align your variables: Make sure both equations are written with variables on the same side (usually $Ax + By = C$).
  2. Eliminate a variable:
    • Get a variable to have the same coefficient number but different signs (e.g., $5x$ and $-5x$).
    • Add the two equations together to cancel that variable out.
    • Solve for the remaining variable.
  3. Substitute and Finish: Take the answer from Step 2 and plug it back into either of the original equations to find the second variable.

Scenario 1: The Perfect Setup

Sometimes, the math gods smile upon us, and the equations are ready to add immediately. Look at this example from the notes:

$$ \begin{cases} 5x + y = 6 \\ -5x + 3y = -22 \end{cases} $$

Notice that we have a positive $5x$ and a negative $5x$. If we add these equations straight down:

  • The $x$'s cancel out ($0$).
  • $y + 3y = 4y$
  • $6 + (-22) = -16$

This leaves us with $4y = -16$. Dividing by 4 gives us $y = -4$. finally, we plug $y$ back in to find $x = 2$. Our solution is $(2, -4)$.

Scenario 2: Creating Opposites

Most of the time, you will need to do a little work to set up the elimination. You might need to multiply one or both equations by a constant to find a Least Common Multiple (LCM).

Let's look at this tougher example from the notes:

$$ \begin{cases} 5x - 2y = -15 \\ 7x + 5y = 18 \end{cases} $$

Here, nothing cancels immediately. Let's choose to eliminate $y$. The coefficients are $-2$ and $5$. The LCM of 2 and 5 is 10.

  • Multiply the top equation by 5 to get $-10y$.
  • Multiply the bottom equation by 2 to get $+10y$.

New System:

$$ \begin{cases} 25x - 10y = -75 \\ 14x + 10y = 36 \end{cases} $$

Now, add them together:

$$ 39x = -39 \Rightarrow x = -1 $$

Substitute $x = -1$ back into the original top equation:

$$ 5(-1) - 2y = -15 \\ -5 - 2y = -15 \\ -2y = -10 \\ y = 5 $$

Final Solution: $(-1, 5)$.

Key Reminders

  • Watch your signs! If you have $4x$ and $4x$, adding them gives $8x$, which doesn't help. You would need to multiply one equation by $-1$ to create $-4x$ so they cancel out.
  • Check your work. Take your final $(x, y)$ coordinate and plug it into both equations to verify it works.

Homework: Please complete page 152, problems #23-28 (all).