Graphing Systems of Equations

Hello Mathletes! Today, we dove into the fascinating world of graphing systems of equations. A system of equations is a set of two or more equations with the same variables. The solution to a system of equations is the point (or points) that satisfy all equations in the system. Graphically, this means finding the point(s) where the lines intersect.

Key Concepts

  • What is a Solution?: A solution to a system of equations is an ordered pair $(x, y)$ that makes all equations in the system true. When we graph the equations, the solution is the point where the lines intersect.
  • Finding the Solution Graphically: Graph each equation on the same coordinate plane. The point where the lines intersect is the solution to the system.
  • Slope-Intercept Form: Remember that the slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This form makes it easy to graph lines.
  • No Solution: If the lines are parallel (same slope, different y-intercepts), the system has no solution. Visually, the lines never intersect. Example: $y = \frac{1}{2}x + 3$ and $y = \frac{1}{2}x - 4$
  • Infinite Solutions: If the lines are the same (same slope and same y-intercept), the system has infinitely many solutions. Visually, the lines overlap completely. Example: $y = -x + 3$ and $2y = -2x + 6$ (which simplifies to $y = -x + 3$)

Examples

Let's look at a couple of examples from our class notes:

Example 1: Finding a Single Solution

Consider the system:

  • $y = x - 3$
  • $y = -x + 5$

By graphing these two lines, we find that they intersect at the point $(4, 1)$. To verify that this is a solution, we substitute $x = 4$ and $y = 1$ into both equations:

  • $1 = 4 - 3 \Rightarrow 1 = 1$ (True)
  • $1 = -4 + 5 \Rightarrow 1 = 1$ (True)

Since the point (4, 1) satisfies both equations, it is indeed the solution to the system.

Example 2: Verifying Solutions

Given the equations:

  • $y = \frac{3}{4}x + 3$
  • $y = 3x - 6$

Do these intersect at the point $(4,6)$? Let's check!

  • $6 = \frac{3}{4}(4) + 3 \Rightarrow 6 = 3 + 3 \Rightarrow 6 = 6$ (True)
  • $6 = 3(4) - 6 \Rightarrow 6 = 12 - 6 \Rightarrow 6 = 6$ (True)

Homework

For homework, please complete the following problems to practice these skills:

  • Page 143: #44, 47
  • Page 155: #1, 2, 5

Remember, practice makes perfect! Don't hesitate to review the video or ask questions in our next class if you need further clarification. Keep up the great work!