Problem 33 - Elementary Algebra Within Reach

Question

When solving a system of linear equations by the method of substitution, how do you recognize that it has infinitely many solutions?

Step-by-Step Solution

Answer

A system of linear equations has infinitely many solutions if, after applying the substitution method, the resulting equation is a true statement involving only numbers (e.g., 0=0). This means the original equations are dependent, representing the same line or plane, and every point on that line or plane is a solution.

Step 1: Understand the system of linear equations

Given a system of linear equations, the first step is to analyze and understand the given equations. The coefficients of the variables within the equations often suggest the relationship between those variables.

Step 2: Apply the substitution method

In substitution method, solve one of the equations for one variable and then substitute that expression into the other equation. This will give a new equation with only one variable.

Step 3: Analyze the results after substitution

Once the substitution has been made, simplify the equation. If the resulting equation is a true statement involving only numbers and no variables (like 0=0), it means the original equations are dependent and represent the same line or coincide with each other. This would suggest an infinite number of solutions as every point on the line is a solution.

Step 4: Conclusion

In conclusion, if the simplified equation from the substitution does not involve variables and is always true, it means that the system of equations has infinitely many solutions. This occurs because the equations represent the same line or planes depending upon the number of variables.

Key Concepts

Substitution Method

The substitution method is a popular technique to solve systems of linear equations. It's a practical approach when one variable in a system can be easily isolated. Here's how you use it:

• First, take one of the equations and solve it for one of the variables. This means you'll express one variable in terms of the other, creating an equation like \( x = 2y + 3 \).
• Next, substitute this expression into the other equation. This effectively replaces the original variable, allowing you to work with just one variable.
• Finally, solve the new equation for the variable. Once you have a solution, substitute it back into the equation you derived initially to find the value of the other variable.

This method is particularly useful in situations where the equations are already structured to facilitate easy isolation of a variable. Remember, patience and practice are key to mastering this technique!

Infinitely Many Solutions

In the context of linear equations, infinitely many solutions occur when any value for the variable will satisfy the equation. When using the substitution method, you might come across a situation where, after substituting and simplifying, you end up with a statement like \( 0 = 0 \).
This indicates that the two equations in the system coincide or represent the same line. Therefore, every point on this line is a solution, leading to infinitely many solutions. The original problem transforms into redundancy because solving it only confirms infinite possibilities.
In practice, recognizing infinitely many solutions often simplifies to spotting when your entire equation reduces down to a true constant statement that does not involve the variables.

Dependent Equations

Dependent equations form a system where all equations are essentially expressions of the same relationship. That means they're essentially multiples of one another, sharing the same graph line or plane.
This occurs when one equation can be derived from another by multiplying by a constant. In the system of linear equations, recognizing dependency means realizing that although you have multiple equations, they don't offer different information. They're redundant.
Dependent equations are the reason behind infinitely many solutions in systems, as they represent overlapping functions. In practical settings, identifying dependent equations early can save time and further computations.