Worked problems: infinite solutions identified by substitution method

📚 Understanding Infinite Solutions in Substitution

In mathematics, when solving a system of equations using the substitution method, identifying infinite solutions is a key concept. It indicates that the two equations are essentially the same, representing the same line if graphed. This guide will help you understand how to recognize and interpret these scenarios.

📜 Historical Context

The study of systems of equations dates back to ancient civilizations, with methods for solving them appearing in Babylonian and Egyptian texts. The formalization of techniques like substitution developed over centuries, becoming a cornerstone of algebraic manipulation.

🔑 Key Principles

• ⚖️ Substitution Process: Involves solving one equation for one variable and substituting that expression into the other equation.
• ♾️ Infinite Solutions: Occur when the substitution results in an identity (a statement that is always true, such as $0 = 0$).
• 📉 Graphical Interpretation: Indicates that the two equations represent the same line, meaning every point on the line is a solution to both equations.

📝 Identifying Infinite Solutions: Worked Examples

Let's explore some examples to illustrate how to identify infinite solutions using the substitution method.

Example 1:

Solve the following system of equations:

Equation 1: $2x + y = 4$

Equation 2: $4x + 2y = 8$

1. Step 1: Solve Equation 1 for $y$:

   $y = 4 - 2x$

2. Step 2: Substitute this expression for $y$ into Equation 2:

   $4x + 2(4 - 2x) = 8$

3. Step 3: Simplify and solve:

   $4x + 8 - 4x = 8$

   $8 = 8$

Since we arrive at the identity $8 = 8$, this indicates that the system has infinite solutions. The two equations represent the same line.

Example 2:

Solve the following system of equations:

Equation 1: $x - y = 1$

Equation 2: $2x - 2y = 2$

1. Step 1: Solve Equation 1 for $x$:

   $x = y + 1$

2. Step 2: Substitute this expression for $x$ into Equation 2:

   $2(y + 1) - 2y = 2$

3. Step 3: Simplify and solve:

   $2y + 2 - 2y = 2$

   $2 = 2$

Again, we arrive at an identity $2 = 2$, indicating infinite solutions.

💡 Tips for Recognizing Infinite Solutions

• 🔍 Look for Multiples: Check if one equation is a multiple of the other. If it is, they represent the same line.
• ✍️ Simplify Equations: Before applying substitution, simplify the equations to see if they are identical.
• ➗ Check for Identities: After substitution, if you arrive at an identity (e.g., $0 = 0$, $5 = 5$), the system has infinite solutions.

🌍 Real-World Applications

While systems with infinite solutions might seem abstract, they appear in various mathematical models where equations are derived from the same underlying relationship. For example, in economic models where multiple equations describe the same market equilibrium, infinite solutions can arise.

🎯 Conclusion

Identifying infinite solutions using the substitution method is a valuable skill in algebra. It demonstrates a deep understanding of how equations relate to each other and provides insights into the nature of mathematical models. By recognizing when equations are dependent, you can avoid unnecessary calculations and interpret the results more effectively.