FTCE Professional Education Practice Exam

What are linear combinations in relation to equations?

1. Using one equation to solve another

2. Graphing two equations together

3. Replacing or combining equations to eliminate a variable

4. None of the above

The correct answer is: Replacing or combining equations to eliminate a variable

Linear combinations in the context of equations refer to the process of combining equations in order to eliminate a variable or to express one equation in terms of another. This concept is particularly useful in solving systems of linear equations. When you take two or more linear equations and manipulate them—by adding, subtracting, or multiplying them by some coefficients—you can create a new equation that can help isolate a variable. This is a fundamental technique in algebra and is often used in methods such as substitution and elimination to find the values of the variables in the equations. For example, if you have two equations representing straight lines, combining them linearly can simplify the problem, making it easier to find the intersection point of the two lines, which corresponds to the solution of the system. This approach leverages the concept of linear independence and manipulation, which is essential for solving complex problems in mathematics systematically.