Problem 35 - Thinking Mathematically

Question

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(3 y=4 x\)

Step-by-Step Solution

Answer

The slope-intercept form of the equation is \(y = \frac{4}{3}x\). The slope is \(\frac{4}{3}\) and the y-intercept is 0.

Step 1: Rewrite in slope-intercept form

Divide both sides of the equation \(3y = 4x\) by 3 to solve for \(y\). The equation becomes \(y = \frac{4}{3}x\), which is now in slope-intercept form.

Step 2: Identify the slope and y-intercept

From the equation \(y = \frac{4}{3}x\), comparison with slope-intercept form indicates the slope \(m = \frac{4}{3}\) and the y-intercept \(c = 0\) (as there is no constant term on the right side of the equation).

Step 3: Graph the line

Start by marking the y-intercept (0) on the y-axis. Then, from that point, move up 4 units and to the right 3 units, as the slope \(\frac{4}{3}\) means rise 4 and run 3. Continue this pattern to draw the line.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form straight lines when graphed. These equations are typically presented in the standard forms such as slope-intercept form or point-slope form. The standard slope-intercept form is written as \( y = mx + c \), where \( m \) represents the slope of the line and \( c \) is the y-intercept. In these equations:

• Every term is either a constant or a product of a constant and a single variable.
• The highest power of any variable is one, meaning no exponents greater than one or radical signs can appear.

Linear equations display a constant rate of change, seen in the uniform distance between the points that lie along the line. This constancy makes them linear, which indicates a straight-line graph. Understanding linear equations serves as a foundation for more advanced mathematics.

Graphing Lines

Graphing lines involves plotting points on a coordinate plane and connecting them to illustrate a linear equation. Here’s how:

• First, identify the equation's y-intercept, which is the constant term \( c \) in the slope-intercept form \( y = mx + c \).
• Plot the y-intercept on the y-axis.
• Next, use the slope \( m \) to find other points on the line. The slope tells you how to move from the y-intercept to the next point, in terms of rise (vertical change) over run (horizontal change).

For example, if the slope is \( \frac{4}{3} \), from the y-intercept, move up 4 units and right 3 units to find the next point. Practice this method repeatedly to enhance graphing skills and precision.

Slope

The slope of a line is a measure of its steepness and direction, usually represented by the letter \( m \) in the slope-intercept equation \( y = mx + c \). Understanding slope involves:

• Recognizing that slope is calculated as \( \frac{\text{rise}}{\text{run}} \), meaning the change in y over the change in x as you move along the line.
• Positive slopes indicate a line that rises as it moves left to right; negative slopes indicate a line that falls.
• A zero slope means the line is horizontal, while undefined slopes imply a vertical line.

The slope \( \frac{4}{3} \) shows that the line rises 4 units for every 3 units it moves right. Understanding this helps predict how changes to the slope will affect the line's graph.

Y-intercept

The y-intercept of a line is where the line crosses the y-axis on a graph, represented by \( c \) in the slope-intercept form \( y = mx + c \). It is where the value of \( x \) is zero, giving a simple point that is easy to plot:

• The y-intercept provides the initial point on a graph before any changes in direction influenced by the slope.
• In the equation \( y = \frac{4}{3}x \), the y-intercept is 0, showing that the line passes through the origin \((0,0)\).
• Knowing the y-intercept simplifies sketching the first key point of the line, a crucial step before extending the line with the slope.

Mastering the identification and utility of the y-intercept aids in more quickly and efficiently graphing lines from equations.