Solving Systems of Equations by Substitution: Part 2

Welcome back to Professor Baker's Math Class! This lesson builds upon our previous work with solving systems of equations using the substitution method. Today, we're tackling systems where neither equation is conveniently solved for a variable at the start. Don't worry; we'll break it down step-by-step!

Key Concepts and Strategies

When neither equation is solved for a variable (like x or y), our first step is to choose one equation and solve it for one of the variables. The goal is to isolate a variable. Here's a breakdown of the process:

  1. Choose an Equation and a Variable: Look for an equation where isolating a variable will be relatively easy. Avoid equations where the variable you want to isolate has a large coefficient or is negative.
  2. Isolate the Variable: Use algebraic manipulations (addition, subtraction, multiplication, division) to get the chosen variable by itself on one side of the equation.
  3. Substitute: Take the expression you found in step 2 and substitute it into the other equation in place of that variable. This will give you a new equation with only one variable.
  4. Solve: Solve the new equation for the remaining variable.
  5. Back-Substitute: Substitute the value you found in step 4 back into either of the original equations (or the expression you found in step 2) to solve for the other variable.
  6. Check Your Solution: Plug both values into both original equations to make sure your solution satisfies both equations! This is a crucial step to avoid errors.

Example Time!

Let's say we have the following system:

$$2x + y = 4$$ $$y = x - 2$$

In this case, the second equation is already solved for y. So $y = x - 2$. Therefore, we will substitute $x-2$ for $y$ in the first equation:

$$2x + (x - 2) = 4$$

Now we can solve for x:

$$3x - 2 = 4$$
$$3x = 6$$
$$x = 2$$

Finally we substitute $x=2$ into $y = x - 2$

$$y = 2 - 2$$
$$y = 0$$

So the final solution is $(2,0)$. Make sure to check your work by plugging the values back into the original equations!

Homework

Practice makes perfect! Complete problems #11-21 (all) on page 152 of your textbook to solidify your understanding of solving systems by substitution.

Keep practicing, and you'll become a substitution superstar! Remember to check your work and don't be afraid to ask questions. Happy solving!