Solving Linear Systems by Graphing: Dependent Systems

In the realm of algebra, linear systems play a crucial role in representing and solving real-world problems involving multiple variables. A linear system is a set of two or more linear equations, each representing a straight line on a graph. These lines can intersect, be parallel, or even coincide, leading to different types of solutions.

One method for solving linear systems is through graphing. This involves plotting the lines represented by the equations and identifying the point(s) of intersection, if any. However, there are cases where the lines do not intersect, indicating a different type of solution. Today, we delve into the concept of dependent systems, where the lines coincide, leading to infinitely many solutions.

Understanding Dependent Systems

A dependent system is a linear system where the equations represent the same line. This means that the lines have the same slope and y-intercept. When graphed, the lines will coincide, meaning they overlap entirely.

Consider the following system of equations:

Equation 1: y = 2x + 3
Equation 2: 2y = 4x + 6

If we simplify Equation 2 by dividing both sides by 2, we get:

y = 2x + 3

Notice that Equation 1 and Equation 2 are now identical. This indicates that the lines represented by these equations will coincide.

Steps to Solve a Dependent System by Graphing

To solve a dependent system by graphing, follow these steps:

Rewrite the equations in slope-intercept form (y = mx + b): This form makes it easier to identify the slope (m) and y-intercept (b) of each equation.
Plot the y-intercept of each equation: The y-intercept is the point where the line crosses the y-axis.
Use the slope to find additional points on each line: The slope represents the change in y over the change in x. For example, a slope of 2 means that for every 1 unit increase in x, y increases by 2 units.
Draw the lines: Connect the points to create the lines represented by the equations.
Observe the lines: If the lines coincide, the system is dependent, and there are infinitely many solutions.

Illustrative Example

Let's solve the following dependent system by graphing:

Equation 1: y = -x + 2
Equation 2: 3y = -3x + 6

Step 1: Rewrite Equation 2 in slope-intercept form:

y = -x + 2

Step 2: Plot the y-intercepts:

Both equations have a y-intercept of 2, so we plot the point (0, 2).

Step 3: Use the slope to find additional points:

The slope of both equations is -1, meaning for every 1 unit increase in x, y decreases by 1 unit.

For example, starting from the y-intercept (0, 2), we can move 1 unit to the right and 1 unit down to find another point on the line, (1, 1).

Step 4: Draw the lines:

Connect the points to create a single line, since both equations represent the same line.

Step 5: Observe the lines:

The lines coincide, indicating that the system is dependent and has infinitely many solutions.

Conclusion

Solving dependent systems by graphing involves recognizing that the equations represent the same line. The lines coincide, leading to infinitely many solutions. By following the steps outlined above, you can effectively identify and solve dependent systems through graphical representation.

Remember that dependent systems are a unique case in linear systems, where the equations are essentially multiples of each other. Understanding these systems is crucial for solving real-world problems and gaining a deeper insight into the relationships between equations and their graphical representations.

Solving Linear Systems by Graphing: Dependent Systems