Problem 69 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-\frac{1}{4} x$$

Step-by-Step Solution

Answer

The plot of the linear equation \(y=-\frac{1}{4} x\) should show a line going downwards from left to right across a graph. The five solutions are formed by substituting selected x-values into the equation and finding the corresponding y-values, which can be displayed as coordinated pairs in a table of values. The five solutions would look similar to [(0, 0), (4, -1), (-4, 1), (8, -2), (-8, 2)].

Step 1: Construct a Table of Values

A table of values is constructed by first selecting values for x, then finding the corresponding y values using the formula \(y=-\frac{1}{4} x\). Five x values are arbitrarily chosen and the corresponding y values are calculated.

Step 2: Calculate Y Values

The y values are calculated for each of the chosen x values by substituting the x value into the equation \(y=-\frac{1}{4} x\). For instance, if x=0, then \(y=-\frac{1}{4} * 0 = 0\). All five x values are substituted into the equation to find their corresponding y values.

Step 3: Present The Solutions

The five solutions are displayed as coordinate points (x, y). For each x value, the corresponding y value is paired to form a coordinate.

Step 4: Plot the Graph

A coordinate system is drawn and the ordered pairs from Step 3 are marked on the system. The points are then joined with a straight line, representing the linear equation which can be used to find other solutions to the equation beyond the five already found.

Key Concepts

Table of Values

A table of values is an essential tool in graphing linear equations. It acts as a bridge between a mathematical formula and a visual representation of that formula on a graph. Constructing a table of values starts with choosing specific input values, which are typically the 'x' values in the context of the equation. For instance, with the equation \(y=-\frac{1}{4}x\), you might select -2, -1, 0, 1, and 2 as your 'x' values.

Once you've decided on the 'x' values, you then apply the equation to find the corresponding 'y' values. This process involves substituting each 'x' value into the equation and solving for 'y'. As an example, substituting '0' into the equation would give a 'y' value of \(0\), because any number multiplied by zero is zero. After calculating each pair, you end up with a set of ordered pairs that act as coordinates which can be plotted on a graph.

Linear Equation Solutions

Solving a linear equation like \(y=-\frac{1}{4}x\), we find the solutions in the form of coordinates. These solutions show the exact points where the equation's graph intersects with the x and y values on a coordinate plane. Importantly, linear equations have an infinite number of solutions since there are countless points lying along the line that the equation represents.

However, for graphing purposes, a handful of specific solutions can often give us enough information to draw an accurate line. In the case of \(y=-\frac{1}{4}x\), by choosing five points, we can establish a reliable pattern. It's also good practice to choose 'x' values that are evenly spaced – including both negative and positive numbers – to get a broad view of the line's behavior across the graph. Remember, the combination of these solutions forms the straight line that characterizes the particular linear equation.

Plotting Coordinates

Once the solutions are found and written as coordinate points (x, y), they are ready to be transferred onto a graph. Plotting coordinates effectively is a vital step in graphing linear equations. Start by drawing a graph with a horizontal number line (x-axis) and vertical number line (y-axis), intersecting at the origin (0,0).

To plot a point, start at the origin. Move horizontally to the 'x' value of the point, then vertically to reach the 'y' value, and mark the spot. Each point should be precise to reflect the true solution of the equation. For example, with a coordinate (2, -0.5), move 2 units right along the x-axis, then 0.5 units down along the y-axis. Perform this action for all your calculated coordinates, then connect them with a straight line. This line is the graphical representation of the linear equation and shows how 'y' changes with respect to 'x'.