FTCE Professional Education Practice Exam

What does the term "linear equations" refer to?

1. Equations that can be graphed as curves

2. Equations with variables raised to powers greater than two

3. Equations that can be represented by straight lines

4. Equations without numerical solutions

The correct answer is: Equations that can be represented by straight lines

The term "linear equations" specifically refers to equations that can be represented by straight lines when graphed on a coordinate plane. These equations typically take the form of \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. The defining characteristic of linear equations is that they involve variables raised to the first power and do not include any products of variables or non-linear transformations (such as squares, square roots, etc.). This structure results in a graph that is a straight line, making linear equations fundamental in algebra and important for understanding relationships between variables in various contexts. In contrast, the other options describe different characteristics that do not apply to linear equations. For instance, equations that can be graphed as curves pertain to non-linear equations, which include quadratic or exponential functions. Equations with variables raised to powers greater than two refer to polynomial equations of higher degrees, which also do not form straight lines. Lastly, equations without numerical solutions could suggest contradictions or non-real solutions, which is not a defining feature of linear equations. Thus, understanding that linear equations yield straight-line graphs is key to their identification and application in mathematics.