Solving Systems of Equations by Graphing

Welcome back to math class! This week, we're focusing on solving systems of equations using graphs. This is a powerful visual method that allows us to find the solutions to multiple equations simultaneously. Remember, a system of linear equations consists of two or more linear equations that have the same variables. The solution to a system of two equations is the ordered pair $(x, y)$ that satisfies both equations. Let's dive in!

Class Notes

Review your class notes, focusing on the key steps for graphing linear equations. The attached PDF provides examples. Key concepts include:

  • Finding Intercepts: The $x$-intercept is the point where the line crosses the x-axis (where $y = 0$), and the $y$-intercept is where the line crosses the y-axis (where $x = 0$).
  • Slope-Intercept Form: Recall that the slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
  • Graphing from a Table: Create a table of values by choosing values for $x$, substituting them into the equation, and solving for $y$. Plot these $(x, y)$ points on the coordinate plane.

Methods for Graphing

  • Graphing Using Intercepts: Find both the $x$ and $y$ intercepts for each equation. Plot these points and draw a line through them. The point where the lines intersect is the solution to the system of equations.
  • Graphing Using Slope-Intercept Form: Convert each equation to $y = mx + b$ form. Plot the y-intercept ($b$) and use the slope ($m$) to find additional points. Draw the lines.
  • Graphing Using a Table: Create a table of $x$ and $y$ values for each equation. Plot the points and draw the lines.

Finding the Solution

Once you've graphed the equations, the solution to the system is the point where the lines intersect. If the lines are parallel and do not intersect, there is no solution. If the lines are the same (overlap completely), there are infinitely many solutions.

Example:

Consider the system of equations:

$$y = x + 1$$ $$y = -x + 3$$

By graphing these two equations, you'll find that they intersect at the point $(1, 2)$. Therefore, the solution to the system is $x = 1$ and $y = 2$. You can verify this by substituting these values back into the original equations.

Homework

Complete the practice pages assigned in the packet. If you're unsure which pages were assigned, please complete both practice pages to ensure you have a solid understanding of the material. The practice pages include problems where you solve systems of equations graphically, which will help solidify your understanding.

Helpful Videos

Here are some video resources to help you:

  • Graphing using Intercepts
  • Graphing using Slope Intercept Form
  • Graphing using a Table
  • Solving Systems by Graphing

Keep practicing, and don't hesitate to ask questions in class! You got this!