Solving Dependent Linear Systems with Substitution

In the realm of algebra, linear systems are a fundamental concept that involves solving for multiple variables in a set of equations. A system is considered dependent when the equations represent the same line, leading to infinitely many solutions. This means that any point on one line also lies on the other, making them essentially identical. This article will guide you through the process of solving dependent linear systems using the substitution method.

Understanding Dependent Systems

Let’s visualize this with an example. Consider the following system of equations:

• Equation 1: 2x + 3y = 6
• Equation 2: 4x + 6y = 12

If you notice, Equation 2 is simply a multiple of Equation 1 (multiplying Equation 1 by 2). This indicates that both equations represent the same line, making the system dependent.

Solving Dependent Systems with Substitution

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. Here’s how it works:

1. Solve for One Variable: Choose one of the equations and solve for one variable in terms of the other. Let’s solve Equation 1 for x:
• 2x + 3y = 6
• 2x = 6 – 3y
• x = (6 – 3y) / 2
2. Substitute: Substitute the expression for x (found in step 1) into the other equation (Equation 2):
• 4x + 6y = 12
• 4[(6 – 3y) / 2] + 6y = 12
3. Simplify and Solve: Simplify the equation and solve for y:
• 2(6 – 3y) + 6y = 12
• 12 – 6y + 6y = 12
• 12 = 12

Notice that the y terms cancel out, leaving us with a true statement (12 = 12). This is a key indicator that the system is dependent.

4. Express Solutions: Since the system is dependent, there are infinitely many solutions. We can express these solutions in terms of one variable. In this case, we can express the solutions as (x, y) where x = (6 – 3y) / 2. This means for any value of y, we can find a corresponding value of x that satisfies both equations.

Example:

Let’s solve another dependent system:

• Equation 1: 3x – 2y = 5
• Equation 2: 9x – 6y = 15

1. Solve for x in Equation 1:
• 3x = 5 + 2y
• x = (5 + 2y) / 3
2. Substitute x into Equation 2:
• 9[(5 + 2y) / 3] – 6y = 15
3. Simplify and solve for y:
• 3(5 + 2y) – 6y = 15
• 15 + 6y – 6y = 15
• 15 = 15
4. Express solutions:
• (x, y) where x = (5 + 2y) / 3

Key Takeaways:

• Dependent systems have infinitely many solutions.
• When solving a dependent system using substitution, you’ll end up with a true statement (e.g., 12 = 12).
• Express the solutions in terms of one variable.

Understanding dependent systems is crucial in linear algebra and helps you interpret the relationship between equations. The substitution method provides a straightforward approach to solving these systems and understanding their infinite solutions.