Introduction
When it comes to solving systems of linear equations, one of the most common methods used is graphing. Graphing allows us to visually represent the equations and find their intersection point, which is the solution to the system. In this article, we will explore the process of solving systems of linear equations by graphing and provide an answer key to help you check your work.
Understanding Systems of Linear Equations
What are systems of linear equations?
Systems of linear equations consist of two or more linear equations that share the same variables. The goal is to find a solution that satisfies all of the equations simultaneously. Each equation represents a line on a graph, and the intersection point of these lines represents the solution to the system.
When is graphing a viable method?
Graphing is a viable method for solving systems of linear equations when the equations are simple and have distinct slopes and intercepts. It allows for a visual understanding of the problem and can be a helpful tool for beginners. However, it may not be the most efficient method for more complex systems.
Steps for Solving Systems of Linear Equations by Graphing
Step 1: Write down the equations
The first step in solving a system of linear equations by graphing is to write down the equations. Make sure they are in standard form, which is Ax + By = C, where A, B, and C are constants. For example, the system:
2x + 3y = 8
4x - y = 5
Step 2: Graph the equations
Next, graph each equation on the same coordinate plane. To do this, convert each equation to slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Plot the y-intercept, and use the slope to find additional points on the line. Repeat this process for each equation.
Step 3: Find the intersection point
The intersection point represents the solution to the system of linear equations. It is the point where the lines intersect on the graph. To find this point, visually locate the intersection on the graph.
Step 4: Check the solution
Once you have found the intersection point, substitute the x and y values into each equation to verify that they satisfy both equations. If the values satisfy both equations, then the solution is correct. If not, recheck your work and make any necessary corrections.
Answer Key for Solving Systems of Linear Equations by Graphing
Example 1
Equation 1: 2x + 3y = 8
Equation 2: 4x - y = 5
Intersection point: (1, 2)
Check:
Equation 1: 2(1) + 3(2) = 8 (True)
Equation 2: 4(1) - 2 = 5 (True)
Solution: (1, 2)
Example 2
Equation 1: 3x - 2y = 7
Equation 2: 2x + y = 4
Intersection point: (3, -2)
Check:
Equation 1: 3(3) - 2(-2) = 7 (True)
Equation 2: 2(3) + (-2) = 4 (True)
Solution: (3, -2)
Example 3
Equation 1: x + y = 5
Equation 2: 2x - y = 1
Intersection point: (2, 3)
Check:
Equation 1: 2 + 3 = 5 (True)
Equation 2: 2(2) - 3 = 1 (True)
Solution: (2, 3)
Conclusion
Solving systems of linear equations by graphing can be a useful method for finding their intersection point and solution. It provides a visual representation of the problem and allows for easy verification. By following the steps outlined in this article and using the provided answer key, you can confidently solve systems of linear equations by graphing.