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Solving Linear Systems by Graphing: Dependent Systems
In the realm of algebra, linear systems encompass a set of two or more equations that share common variables. Solving such systems involves finding the values of the variables that satisfy all equations simultaneously. One powerful technique for tackling linear systems is graphing. This method offers a visual representation of the solutions, making it intuitive and insightful.
When we graph a linear system, each equation translates into a straight line. The point where these lines intersect represents the solution to the system. This point satisfies both equations, signifying the values of the variables that make both equations true.
Understanding Dependent Systems
Dependent systems are a special type of linear system where the equations represent the same line. This means the lines coincide, sharing all their points in common. Consequently, dependent systems have infinitely many solutions.
How to Solve Dependent Systems by Graphing
1. **Graph each equation:** Start by graphing each equation in the system. You can use any method you prefer, such as slope-intercept form (y = mx + b) or plotting points.
2. **Observe the lines:** Carefully examine the lines you've graphed. If the lines coincide (overlap completely), you have a dependent system.
3. **Identify the solution:** Since the lines coincide, every point on the line represents a solution to the system. Therefore, the solution is infinite, and you can express it as a general solution that represents all the points on the line.
Example:
Let's consider the following system:
Equation 1: 2x + 3y = 6
Equation 2: 4x + 6y = 12
To solve this system by graphing, we'll first rewrite each equation in slope-intercept form:
Equation 1: y = (-2/3)x + 2
Equation 2: y = (-2/3)x + 2
Notice that both equations have the same slope (-2/3) and the same y-intercept (2). This indicates that the lines are identical. When we graph these equations, we'll find that they overlap completely.
Therefore, the system is dependent, and it has infinitely many solutions. The solution can be expressed as a general solution, such as:
x = t
y = (-2/3)t + 2
Where 't' represents any real number.
Conclusion:
Solving dependent systems by graphing is a straightforward process. By visualizing the lines represented by the equations, we can quickly determine if they coincide. If they do, we know the system is dependent and has infinitely many solutions. This method provides a clear and intuitive understanding of dependent systems and their solutions.