Solving Systems of Equations by Graphing
Today, we began exploring how to solve systems of equations by graphing. This is a fundamental concept in algebra, and we'll continue to build upon it in our next class on Tuesday. Let's review what we covered and what you need to practice.
Key Concepts
A system of linear equations involves two or more linear equations with the same variables. The solution to such a system is an ordered pair $(x, y)$ that satisfies all equations simultaneously. Graphing provides a visual way to find this solution. When we graph two equations, the point where the lines intersect represents the solution because that point lies on both lines, hence satisfying both equations.
To solve a system by graphing, follow these steps:
- Graph each equation on the same coordinate plane. You can graph a line by finding the x and y intercepts.
- Identify the point of intersection. The coordinates of this point represent the solution to the system.
- Check your solution. Substitute the x and y values of the intersection point into both original equations to verify that they hold true.
For example, consider the system:
$$ -x + y = 3 $$ $$ 2x + y = 6 $$By graphing these equations, we can see that they intersect at the point (1, 4). Substituting these values into the equations:
For the first equation: $$-(1) + 4 = 3$$ which simplifies to $$3 = 3$$.
For the second equation: $$2(1) + 4 = 6$$ which simplifies to $$6 = 6$$.
Since (1, 4) satisfies both equations, it is the solution to the system.
Homework
- Reflect: Complete questions 1a-1c in the packet. These questions will help you think critically about the concepts we covered in class.
- Practice Problems: Do Practice #1 in the packet to get more experience solving systems of equations by graphing.
- Textbook: Complete problems #11-19 all on page 142 in the textbook.
Remember, practice makes perfect! Don't hesitate to ask questions if you're feeling stuck. I'm here to help you succeed!