Solving Systems of Equations by Substitution - November 13, 2013

Welcome back to Professor Baker's Math Class! Today, we're diving into solving systems of equations using the substitution method. This is a powerful technique that allows us to find the values of two or more variables when we have multiple equations.

What is Substitution?

The substitution method involves solving one equation for one variable and then substituting that expression into another equation. This results in a single equation with one variable, which we can then solve. After solving for one variable, we can substitute that value back into either of the original equations to find the value of the other variable. Here's a breakdown of the steps involved:

  1. Solve for one variable: Choose one of the equations and solve it for one variable in terms of the other. For example, solve for $y$ in terms of $x$, or vice versa.
  2. Substitute: Substitute the expression you found in Step 1 into the other equation. This will give you an equation with only one variable.
  3. Solve for the remaining variable: Solve the equation you obtained in Step 2 for the remaining variable.
  4. Substitute back: Substitute the value you found in Step 3 back into either of the original equations (or the expression from Step 1) to find the value of the other variable.
  5. Check your solution: Substitute both values into both original equations to ensure they are true.

Example

Let's solve the following system of equations:

$$y = x + 3$$

$$y = 3x - 4$$

Since both equations are already solved for $y$, we can set them equal to each other:

$$x + 3 = 3x - 4$$

Now, solve for $x$:

$$3 + 4 = 3x - x$$

$$7 = 2x$$

$$x = \frac{7}{2}$$

Substitute $x = \frac{7}{2}$ back into the first equation to find $y$:

$$y = \frac{7}{2} + 3$$

$$y = \frac{7}{2} + \frac{6}{2}$$

$$y = \frac{13}{2}$$

So the solution is $x = \frac{7}{2}$ and $y = \frac{13}{2}$ or $(\frac{7}{2}, \frac{13}{2})$.

Homework

Complete all the assigned problems to practice what you've learned! Problems 5 and 6 are required to get credit, so make sure you give them a good attempt. Don't be afraid to ask questions if you get stuck.

Here is a video that can help you solve systems of equations with substitution: