Problem 37 - College Algebra

Question

For each of the following exercises, construct a table and graph the equation by plotting at least three points. $$2 y=x+3$$

Step-by-Step Solution

Answer

Graph the equation by plotting points from the table and connecting them.

Step 1: Convert the equation to solve for y

To make plotting easier, we should rewrite the equation in the form of \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Starting with \(2y = x + 3\), divide everything by 2: \(y = \frac{x}{2} + \frac{3}{2}\).

Step 2: Create a table of values

Choose at least three values for \(x\) to substitute into the equation \(y = \frac{x}{2} + \frac{3}{2}\) and find the corresponding values of \(y\). For example, if \(x = 0\), \(y = \frac{0}{2} + \frac{3}{2} = \frac{3}{2}\), if \(x = 2\), \(y = \frac{2}{2} + \frac{3}{2} = 2\), and if \(x = -2\), \(y = \frac{-2}{2} + \frac{3}{2} = \frac{1}{2}\).

Step 3: Plot the points on a graph

Plot each point obtained from the table onto a graph. Using the points \((0, \frac{3}{2})\), \((2, 2)\), and \((-2, \frac{1}{2})\), place a dot at each position in an appropriate coordinate system.

Step 4: Draw the line through the points

Connect the points with a straight line, which will represent the graph of the equation \(y = \frac{x}{2} + \frac{3}{2}\). Ensure that the line extends in both directions beyond the plotted points, as it continues infinitely.

Key Concepts

slope-intercept form

The slope-intercept form is one of the most common ways to express a linear equation. It is written as \(y = mx + c\), where \(m\) represents the slope of the line, and \(c\) represents the y-intercept. The slope \(m\) determines how steep the line is, and it tells us how much \(y\) changes for each unit change in \(x\).
For example, if \(m=\frac{1}{2}\), it means for every increase of 1 in \(x\), \(y\) increases by \(\frac{1}{2}\).
The y-intercept \(c\) is where the line crosses the y-axis. It indicates the value of \(y\) when \(x=0\).
This form is particularly useful for quickly graphing linear equations because it directly shows both the slope and the intercept, allowing us to understand and visualize the line's direction and starting point.

coordinate system

The coordinate system is a framework used to locate points on a plane. It consists of two perpendicular lines, usually drawn on graph paper or digital representations, called the x-axis (horizontal) and y-axis (vertical).
These axes divide the plane into four quadrants, each with its own characteristics, determined by the sign (positive or negative) of the x and y coordinates.

• The point where the axes intersect is called the origin, designated as (0, 0).
• Coordinates are written as pairs, (x, y), indicating a point's location relative to the origin.

By using this system, we can plot points, draw lines, and observe relationships between variables easily and systematically. Whether working with physics, mathematics, or other fields, the coordinate system is a crucial tool for visual representation and analysis.

plotting points

Plotting points is the process of placing dots on a graph to represent values from ordered pairs, each consisting of an x-coordinate and a y-coordinate.
To plot a point like (2, 3), start at the origin (0, 0), move 2 units along the x-axis to the right, and then move 3 units up parallel to the y-axis.
This dot is the visual representation of the point on your graph.

• Repeat this process for different points to form part of a line or curve.
• Once points are plotted, they can be connected to show trends or relationships.

In the context of linear equations, you'll often pick and plot three points to ensure accuracy in drawing the line.
By following the equation in slope-intercept form, you can easily determine points and create an accurate graph that clearly shows the line's behavior across different values.