Problem 43 - College Algebra

Question

Give the slope and y-intercept of each line whose equation is given. Then graph the line. $$y=\frac{3}{4} x-2$$

Step-by-Step Solution

Answer

The slope of the line is \(\frac{3}{4}\) and the y-intercept is \(-2\). Limitations in text representation don't allow for the real-life graphing portion of this solution, one must draw this on paper or use graphing software to visualize the line.

Step 1: Identify the slope and y-intercept

The given equation is \(y = \frac{3}{4}x - 2\), it's already in the form \(y = mx + b\). Here, the coefficient of \(x\) is the slope \(m = \frac{3}{4}\), and the constant term \(-2\) is the y-intercept \(b\).

Step 2: Plotting the y-intercept

The y-intercept is \(-2\), which represents the point on the y-axis where the line crosses. So, the first point to plot on the graph is \((0, -2)\). Draw a small dot at this point on the y-axis.

Step 3: Plotting more points using the slope

The slope being \(\frac{3}{4}\) means 'rise 3 and run 4'. From the y-intercept, count 3 units upward (positive because it's 3), and 4 units rightward (positive because it's 4). Each of these new points is also on the line.

Step 4: Draw the line

Using a ruler, neatly draw a straight line that goes through all the plotted points. The line represents all the solution of the given equation.

Key Concepts

Understanding the Slope

The slope of a line, often represented by the letter *m* in the equation of a line, tells us how steep the line is. In a linear equation written in slope-intercept form, i.e., \(y = mx + b\), the slope is the coefficient of \(x\).

In the example given, the equation \(y = \frac{3}{4}x - 2\) has a slope of \(\frac{3}{4}\). This fraction indicates that for every 4 units you move horizontally (to the right), the line moves up 3 units vertically. This is called "rise over run."

• If the slope is positive, like \(\frac{3}{4}\), the line goes upwards as you move from left to right.
• If the slope were negative, the line would slope downwards.
• A larger slope value means a steeper line, whereas a smaller slope means a gentler incline.

Recognizing and working with the slope is essential for plotting lines and understanding how changes in values affect the angle and direction of your line.

Identifying the Y-Intercept

In the slope-intercept form of a linear equation \(y = mx + b\), the *y-intercept* is the term \(b\). It refers to the point where the line crosses the y-axis, which occurs when \(x = 0\).

For the equation \(y = \frac{3}{4}x - 2\), the y-intercept is \(-2\). This means that the line will intersect the y-axis at the point \((0, -2)\). It's crucial to plot this point as your first step when graphing a line because it serves as the starting position.

• A positive y-intercept means the line crosses above the origin.
• A negative y-intercept, like in our example, indicates it crosses below the origin.

By knowing the y-intercept, you gain a clear initial point to begin graphing, which simplifies the drawing of accurate lines on a coordinate plane.

Tips for Graphing Lines

Graphing lines involves transforming an equation into a visual representation on a coordinate plane. Here's a step by step approach using the provided equation \(y = \frac{3}{4}x - 2\):

• Start with the y-intercept: Identify the y-intercept from the equation (\(-2\)) and plot this on the y-axis at \((0, -2)\).
• Use the slope for direction: From the y-intercept, use the slope \(\frac{3}{4}\). Move 3 units up and 4 units to the right to find another point on the line, e.g., reaching the point \((4, 1)\).
• Connect the dots: After plotting these points, draw a straight line through them using a ruler. Ensure the line extends in both directions, covering the entire plane.

Graphing lines using slope and y-intercept ensures that the depiction is based on the precise mathematical relationship expressed in the equation, creating a clear visual representation of the line's position and direction.