91Ó°ÊÓ

Linear equations are equations of the first degree, meaning they have no exponents higher than one. They usually appear in two variables, such as \( x \) and \( y \), and take on the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.

In our exercise, the linear equation is in terms of \( y \) and \( x \). This specific equation, \( y = \frac{1}{4}x \), is a simple linear equation showing a direct relationship between \( y \) and \( x \). In this equation, the coefficient \( \frac{1}{4} \) is the slope.

The line is drawn across the coordinate plane and it has a special attribute in this case as it will form the boundary for an inequality graph.

The coordinate plane, or Cartesian plane, is a two-dimensional surface where you can plot points using two numbers called coordinates. These coordinates, traditionally \( x \) and \( y \), show the position of a point in relation to the horizontal axis (x-axis) and the vertical axis (y-axis).

Each point on the plane is identified by a pair \((x, y)\) that represents its distance from the x-axis and y-axis. By using a grid pattern, the coordinate plane helps you visualize linear equations and inequalities.

In our case, plotting the line \( y = \frac{1}{4}x \) involves marking points on the plane to help identify where the boundary line sits and how to proceed with shading the correct region underneath it.

The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is particularly useful because it clearly displays the slope and starting point of the line on the y-axis.

In the exercise, the equation \( y = \frac{1}{4}x \) is already in slope-intercept form where the slope \( m \) is \( \frac{1}{4} \), and the y-intercept \( b \) is 0.

This means the line will start at the origin (0,0) and with every 1 step horizontally on the x-axis, it will rise \( \frac{1}{4} \) units vertically. This approach makes graphing the line fairly straightforward, helping to create the boundary needed to shade the appropriate region related to the inequality.

Shading regions in the context of graphing inequalities involves identifying which part of the coordinate plane satisfies the inequality. Depending on the inequality sign, such as \( \leq \) or \( \geq \), the shading can be above or below the line.

For the inequality \( y \leq \frac{1}{4}x \), we shade the region below the line \( y = \frac{1}{4}x \). This shaded area includes all the points where \( y \) is less than or equal to \( \frac{1}{4}x \).

The solid line itself is part of the solution because the inequality includes an equivalence ('='). The shaded region graph helps clearly illustrate which values of \( x \) and \( y \) satisfy the inequality, making it easier for one to understand the solution conceptually.