Problem 77 - 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. $$y=x+\frac{1}{2}$$

Step-by-Step Solution

Answer

The graph of the equation \(y = x + \frac{1}{2}\) is a straight line that crosses the y-axis at \(\frac{1}{2}\) and slopes upwards. Five solutions for the equation could be: (-2, -1.5), (-1, -0.5), (0, 0.5), (1, 1.5), and (2, 2.5)

Step 1: Identify the slope and y-intercept

In the given linear equation \(y = x + \frac{1}{2}\), identify the slope which is \(1\), and the y-intercept which is \(\frac{1}{2}\). This is because the equation is in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept

Step 2: Plot the y-intercept

Start the graph by plotting the y-intercept, which is at point (0, \(\frac{1}{2}\)). This is the point where the line crosses the y-axis

Step 3: Use the slope to find other points

The slope being 1 means for every unit increase in \(x\), \(y\) increases by the same amount. So, choose a few \(x\)-values (e.g., -2, -1, 0, 1, 2) and use the slope to identify the corresponding \(y\)-values.

Step 4: Draw the line

After plotting points for the selected \(x\)-values, connect the points to draw the line that represents this linear equation.

Step 5: Find the solution

To find solutions to the equation, list all the \(x\), \(y\) pairs from the graph or calculated while determining points to plot. As per requested, list at least five solutions.

Key Concepts

Slope-intercept form

One of the most straightforward ways to represent a linear equation is through the slope-intercept form. This form is given by the equation \(y = mx + c\). Here, \(m\) represents the slope of the line, while \(c\) denotes the y-intercept.

• The slope \(m\) tells us how steep the line is.
• The y-intercept \(c\) is the point where the line crosses the y-axis.

This form makes graphing linear equations much easier because you can quickly identify these two key features. To use the slope-intercept form effectively, identify \(m\) and \(c\) right away. For example, in the equation \(y = x + \frac{1}{2}\), the slope \(m\) is \(1\) and the y-intercept \(c\) is \(\frac{1}{2}\).

Y-intercept

The y-intercept is a crucial part of understanding linear equations as it provides a fixed starting point for the graph. In the equation \(y = mx + c\), the y-intercept is \(c\). This is the value of \(y\) when \(x\) is zero.To visualize this, imagine starting to draw your graph on the y-axis. For instance, with our equation \(y = x + \frac{1}{2}\), the point on the y-axis where you start plotting is (0, \(\frac{1}{2}\)). This location is the anchor from which your line begins.Understanding the y-intercept assists you in quickly establishing where the line crosses the y-axis, providing a reference point for plotting additional points.

Slope

The slope of a line is a measure of its steepness and direction. In the slope-intercept form, represented as \(m\), it indicates how much \(y\) changes for a given change in \(x\).

• If \(m\) is positive, the line goes upwards from left to right.
• If \(m\) is negative, the line slopes downwards.
• For a slope of zero, the line is horizontal, meaning no change in \(y\) as \(x\) changes.

For the equation \(y = x + \frac{1}{2}\), the slope \(m\) is \(1\). This means for each increase or decrease in \(x\) by 1 unit, \(y\) increases or decreases by the same amount. Thus, if you chose an \(x\) value 2 units away from a previous one, the \(y\) value would also move by 2 units.

Linear functions

Linear functions are all about constant rates and straight lines. They exhibit a direct relationship between \(x\) and \(y\), displaying equal changes in both variables. In their most basic form, these functions can be written as \(y = mx + c\), resembling a straight line when graphed.Linear functions are simple but powerful, often serving as introductions to larger algebraic concepts. Consider our exemplary equation, \(y = x + \frac{1}{2}\), which embodies these principles:

• Its graph is a straight line, confirming the linear nature.
• The slope emphasizes a constant rate, represented as a ratio of the change in \(y\) over the change in \(x\).
• The y-intercept illustrates a starting value from which other y-values are derived.

Understanding linear functions forms a foundation for studying more complex mathematical relationships.