Question

State whether or not the given equation is linear. $$\sqrt{5} y+\pi x=-1$$

Short answer

Yes, the equation is linear.

Step-by-step solution

Identify the Standard Form of a Linear Equation

A linear equation in two variables generally has the form \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants, \( x \) and \( y \) are variables, and the variables are only to the first power (not squared or inside functions like square roots, etc.).

Examine the Given Equation

The given equation is \( \sqrt{5}y + \pi x = -1 \). Here, the coefficients of \( y \) and \( x \) are \( \sqrt{5} \) and \( \pi \) respectively, which are constants as they do not change values like \( x \) or \( y \). The right side is also a constant \(-1\).

Check for Linear Terms

Inspect each term to ensure all variables are raised to the first power. In the equation \( \sqrt{5}y + \pi x = -1 \), both \( y \) and \( x \) appear to the first power, with no variable being squared, under a square root with the variable, or included in any other non-linear operations.

Conclude Whether the Equation is Linear

Since all terms appropriated to the variables are linear (i.e., the variables \( x \) and \( y \) are both to the first power), and there are no multiplicative products of different variables, the equation matches the standard form of a linear equation. Therefore, the given equation is linear.

Key concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all mathematics and involves everything from solving elementary equations to studying abstractions. In basic algebra, we often encounter linear equations, which are key to understanding how variables relate to one another. Linear equations form the foundation of many algebraic applications. These equations model relationships in which changes in one variable produce directly proportional changes in another. They can appear straightforward at times, yet they serve as building blocks for more complex mathematical ideas. The underlying principle is to find the value of one variable when the other variable(s) is given.

Variables

Variables are fundamental to algebra and act as placeholders for numbers or quantities that can change or vary. They are typically represented by letters such as \( x \) or \( y \). In linear equations, variables are crucial as they represent unknowns that we aim to solve. For example, in the equation \( \sqrt{5}y + \pi x = -1 \), \( x \) and \( y \) are variables.
Variables allow algebra to explore and solve real-world problems. They enable us to generalize mathematical statements, making it easier to model relationships and prove results.
Understanding how to manipulate and operate with variables lays a critical foundation for solving equations and developing more complex mathematical theories.

Coefficients

Coefficients are the numerical or constant factors attached to variables in an equation. They provide the scalar value for the variable, influencing the slope or rate of change in the equation. In our example, \( \sqrt{5} \) is the coefficient of \( y \) and \( \pi \) is the coefficient of \( x \). Although these coefficients look unconventional due to their irrational nature, they remain constants and do not affect the linearity of the equation.

• Coefficients allow us to weigh the importance or impact of different variables within the equation.
• They adjust the variable's value, showing how much the variable contributes to achieving equality in the equation.

Recognizing how coefficients affect variables is pivotal in analyzing and graphing linear equations, as they play a central role in determining the slope of the line.

Standard Form

Standard form is a way of writing linear equations so that it's easier to analyze and interpret them. It usually follows the structure \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants, and \( x \) and \( y \) are variables. This form is particularly useful when solving systems of equations and performing calculations with multiple equations. The given equation \( \sqrt{5}y + \pi x = -1 \) aligns with this format, demonstrating it fits into the standard form of a linear equation.
Linear equations in standard form have broad applications in graphing, solving simultaneous equations, and mathematical modeling.
Understanding how to convert different forms into the standard form is a valuable skill that simplifies solving and interpreting complex mathematical problems.