Question

Name the following form of the equation of a line: \(A x+B y=C\).

Short answer

The form of the line equation \( A x + B y = C \) is the standard form.

Step-by-step solution

Identifying the Given Equation

Look at the given equation \( A x + B y = C \). It's a linear equation because the two variables x and y are to the first power, they are not squared or cubed, and there is no multiplication between variables.

Comparing with Standard Form

Recall the standard form of the equation of a line which is usually written as \( Ax+By=C \), where A and B are not both equal to zero. We quickly see that the given equation matches this form.

Naming the Equation Form

Therefore, we conclude that the form of the given line equation \( A x + B y = C \) is the standard form of the equation of a line.

Key concepts

Linear Equation

When we talk about linear equations in algebra, we refer to mathematical expressions that depict a straight line when plotted on a graph. Fundamentally, a linear equation represents a constant relationship between two variables – typically 'x' and 'y' – wherein each variable is raised to the first power. That means no exponents higher than one, and no multiplying the variables together. A classic example is the slope-intercept form, which looks like \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept.

The simplicity of linear equations makes them a fundamental part of algebra and essential for students to understand. Their applications stretch far and wide, from calculating finances to understanding science principles. In solving these equations, variables are isolated to find their values. It's like a balancing act: what you do to one side of the equation, you must also do to the other, keeping the equation equal and balanced.

Variables in Algebra

Algebra is a form of mathematics that utilizes letters known as variables to represent unknown numbers. Variables can be any letter of the alphabet, but 'x' and 'y' are the most frequently used. They are placeholders, waiting to be replaced with specific values that make the equation true.

In an equation, each variable represents a dimension. For example, in a one-dimensional linear equation, only one variable like 'x' is necessary. However, with two dimensions, both 'x' and 'y' variables are employed, allowing us to plot the equation on a coordinate plane. It's essential to distinguish independent and dependent variables in a function or equation. Typically, 'x' is independent, meaning it can take on any value, and 'y' is dependent, as its value relies on 'x'. Understanding variables is crucial because algebra teaches us the language of patterns, and being fluent in this language opens the door to solving real-world problems.

Equation of a Line

An equation of a line provides a recipe for drawing that line on a coordinate plane. Every point on the line satisfies the equation, and conversely, every solution to the equation is a point on the line. This duality is what makes line equations so powerful. Various forms of the equation of a line exist, with each highlighting different aspects of the line such as slope, intercepts, or normal vector. One of the most important forms is the standard form, \( Ax+By=C \), praised for its simplicity and reliability.

The standard form can readily reveal the x-intercept \( x = \frac{C}{A} \) and y-intercept \( y = \frac{C}{B} \) when either variable is set to zero, respectively. In education, understanding the standard form is emphasized because of its utility in graphing and its use in systems of equations. Students should also be aware of how to convert between different forms, such as from slope-intercept to standard form, a skill that reinforces the understanding of the relationships between algebraic expressions and graphical representations.