Problem 42 - College Algebra with Modeling and Visualization

Question

When using elimination and substitution, explain how to recognize a system of linear equations that has infinitely many solutions.

Step-by-Step Solution

Answer

A system has infinitely many solutions when the equations are equivalent, resulting in an identity like 0 = 0.

Step 1: Understanding Infinitely Many Solutions

A system of linear equations has infinitely many solutions when the equations are dependent, meaning they represent the same line. This occurs when the equations are multiples of each other.

Step 2: Using Elimination Method

To check for infinitely many solutions using elimination, attempt to eliminate the same variable from both equations. If you find an identity (like 0 = 0) after elimination, the system has infinitely many solutions.

Step 3: Using Substitution Method

To check using substitution, solve one equation for a variable and substitute it into the other equation. If you end up with an identity or an equation that is always true (like 0 = 0), then the system has infinitely many solutions.

Step 4: Example Breakdown

For instance, consider the system: \( 2x + 3y = 5 \) and \( 4x + 6y = 10 \). Multiply the first equation by 2 to get \( 4x + 6y = 10 \), which is exactly the second equation, indicating the system has infinitely many solutions.

Key Concepts

elimination method

The elimination method is a powerful way to solve systems of linear equations. It's ideal for situations where you can easily add or subtract the equations to eliminate one of the variables.
This method involves aligning two equations in such a way that one of the variables cancels out when the equations are added or subtracted. For example, if you have equations like \(2x + 3y = 5\) and \(4x + 6y = 10\), you might notice that multiplying the first equation by 2 will make the coefficients of \(x\) in both equations equal.
Subtracting or adding the equations can then be used to eliminate \(x\). When you find that the result is an identity like \(0 = 0\) after this elimination step, it confirms that the system has infinitely many solutions. This means the equations are essentially the same, just written differently, representing a situation where every solution to one is a solution to both.

substitution method

The substitution method offers another efficient way to address systems of linear equations, especially when one equation is already solved for a variable, or it can easily be manipulated to do so. This method involves isolating one variable in one equation and then substituting this expression into the other equation.
For example, consider the system \(y = \frac{5-2x}{3}\) and substituting it into \(4x + 6y = 10\). By solving for \(y\) in the first equation, and then replacing \(y\) in the second, you can determine if the resulting expression is always true. If it resolves to an identity such as \(0 = 0\), this indicates infinitely many solutions.
Essentially, it's like discovering that both equations define the same line by finding an infinite set of points that satisfy both equations simultaneously.

infinite solutions

Identifying a system with infinite solutions involves recognizing when equations are dependent. Dependent equations are really just different versions of the same line. This scenario results in an infinite number of solutions because there are countless points along the line that satisfy the system.
This can occur through either the elimination or substitution method. In the elimination method, if eliminating a variable leads to an identity like \(0 = 0\), it confirms infinite solutions. During substitution, if substituting for a variable yields a statement that is inherently true for all values, such as \(0 = 0\), you again have infinite solutions.
For a tangible example, consider any two equations like \(2x + 3y = 5\) and \(4x + 6y = 10\). By manipulating these equations, multiplying or altering them, you find they represent the same line, confirming infinite solutions. This is because every solution on that line satisfies both equations perfectly.