Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$5 y=20$$

Step-by-Step Solution

Answer

The graph of the equation \(5 y = 20\) is a straight line parallel to the x-axis at y = 4.

Step 1: Simplify the equation

To simplify the equation \(5y = 20\), divide both sides by 5. This gives the simplified equation \(y = 4\). This is a straight line parallel to the x axis, with y coordinate as 4.

Step 2: Plot y=4 on a cartesian plane

On the cartesian plane (also called a grid), mark y=4. Draw a straight line parallel to the x-axis passing through the point where y=4. This represents the equation \(y = 4\). All the points on this line have y-coordinate as 4.

Key Concepts

Cartesian Plane

The Cartesian plane, also known as the coordinate plane, is a two-dimensional surface where you can plot points, lines, and curves to represent algebraic equations visually. It consists of two perpendicular lines: the horizontal axis (x-axis) and the vertical axis (y-axis). These axes intersect at the origin, which is the point (0, 0).

Each point on this plane can be identified by a pair of numbers, \(x, y\), known as coordinates. The x-coordinate indicates the position along the x-axis, while the y-coordinate shows the position along the y-axis.

Understanding how to navigate and plot points on the Cartesian plane is essential for graphing equations and analyzing mathematical relationships visually. This system originated from the French mathematician René Descartes, hence the name Cartesian.

Simplifying Equations

Simplifying equations is a crucial skill in algebra that involves reducing equations to their simplest form. This often makes equations easier to understand and solve, especially when they are more complex.

For linear equations, simplifying typically involves combining like terms, removing unnecessary components, or applying operations that make the equation more straightforward. In the example \(5y = 20\), simplifying involves dividing both sides by 5, which gives \(y = 4\). This process helps transform the equation into a standard form, allowing an easier interpretation or graphing.

Simple equations usually have one variable solved in terms of constants, like \(y = 4\), making subsequent steps, such as graphing, more intuitive.

Graphing Linear Equations

Graphing linear equations involves plotting straight lines on the Cartesian plane. Linear equations, by definition, describe lines, usually in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. However, for horizontal or vertical lines, the equations can look simpler, such as \(y = c\) or \(x = c\).

In our example, \(y = 4\) is the equation of a horizontal line where every point has a y-coordinate of 4. To graph this, you plot a point at \((0, 4)\) on the Cartesian plane and draw a straight line parallel to the x-axis.

You can broaden this approach to any linear equation once you simplify it. Knowing the type of line your equation signifies helps you accurately trace its path on the graph. The visual representation aids in better understanding the relationship between variables in the equation.