Welcome back to Professor Baker's Math Class! In our previous lessons, we learned how to solve systems of equations by graphing. While graphing is visual, it isn't always precise. Today, we are exploring an algebraic approach: Solving Systems by Substitution.

This method is particularly powerful when one of your equations has a variable with a coefficient of 1 or -1. Below is the step-by-step process and the class notes from the lecture.

The 3-Step Substitution Process

As outlined in the lecture notes, there is a logical flow to finding the solution:

  1. Isolate: Solve one of the equations for a specific variable (get $x$ or $y$ by itself). Tip: Choose the variable that is easiest to isolate!
  2. Substitute: Take the expression from Step 1 and plug it into the other equation. Solve for the remaining variable.
  3. Solve: Substitute the answer from Step 2 back into the equation from Step 1 to find the second coordinate.

Example Walkthrough

Let's look at an example from the class notes to see this in action.

System:
$$3x + 4y = -4$$ $$x + 2y = 2$$

Step 1: We choose the second equation because $x$ is easy to get by itself.
$$x = 2 - 2y$$

Step 2: Substitute $(2 - 2y)$ wherever you see $x$ in the first equation:
$$3(2 - 2y) + 4y = -4$$
Distribute and solve:
$$6 - 6y + 4y = -4$$
$$6 - 2y = -4$$
$$-2y = -10$$
$$y = 5$$

Step 3: Plug $y = 5$ back into our Step 1 equation:
$$x = 2 - 2(5)$$
$$x = 2 - 10$$
$$x = -8$$

Final Answer: The lines intersect at $(-8, 5)$.

Special Cases

Sometimes, the variables will completely disappear while you are solving. Don't panic! This tells you something specific about the lines:

  • Infinite Solutions (Same Line): If the variables cancel out and you are left with a true statement (e.g., $-6 = -6$ or $-34 = -34$), the lines are identical. Every point on the line is a solution.
  • No Solution (Parallel Lines): If the variables cancel out and you are left with a false statement (e.g., $6 = 4$ or $12 = 10$), the lines are parallel and will never touch.

Class Assignment

To practice these skills, please complete the following assignment from the textbook:

  • Page 152: Problems #11-22 (all)
  • Note: Pay extra attention to problems #14, 15, 16, 17, and 20 as they involve fractions!

Download the full PDF class notes attached to this post to see more worked-out examples, including the special cases. Keep practicing, and I'll see you in class!