Problem 26 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ y=-\frac{3}{4} x-1 $$

Step-by-Step Solution

Answer

Graph the line starting at (0, -1) using the slope of -3/4 to find another point.

Step 1: Understand the Equation

The given equation is in slope-intercept form, which is \( y = mx + b \). Here, \( m = -\frac{3}{4} \) (the slope), and \( b = -1 \) (the y-intercept). This means the line crosses the y-axis at -1 and has a slope of -3/4.

Step 2: Identify the Y-Intercept

The y-intercept of the equation is \( b = -1 \). This means the graph will cross the y-axis at the point \((0, -1)\). This is your starting point for graphing.

Step 3: Apply the Slope

The slope \( m = -\frac{3}{4} \) means that for every 4 units you move to the right (positive x-direction), you move down 3 units (negative y-direction). From the y-intercept \((0, -1)\), move 4 units to the right and 3 units down to plot a new point at \((4, -4)\).

Step 4: Plot the Points and Draw the Line

On the graph, plot the y-intercept \((0, -1)\) and the point you calculated, \((4, -4)\). Draw a straight line through these two points, extending in both directions to cover the graph.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations so that they are easy to graph. This form is written as \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating its steepness and direction
• \( b \) is the y-intercept, indicating where the line crosses the y-axis

Using slope-intercept form makes it straightforward to understand the overall behavior of the line. For example, if \( m \) is positive, the line slants upwards from left to right. If it's negative, as in our example, the line slants downwards. The value of \( b \) provides a clear starting point: where the line hits the y-axis. This makes plotting and visualizing the graph faster and more intuitive.

Plotting Points

Plotting points on a graph helps us to visualize linear equations. We start with the y-intercept as our first point. From there, we use the slope to find additional points along the line.
Let's consider the equation from our example: \( y = -\frac{3}{4} x - 1 \). We first plot the y-intercept at (0, -1). Then, with a slope of \(-\frac{3}{4}\):

• From the y-intercept, move 4 units to the right
• Then move 3 units down

This gives us a new point at (4, -4). Remember, plotting doesn't stop here - it's important to continue the pattern in both directions to ensure the line covers the whole graph. The key is consistency and following the slope meticulously.

Y-Intercept

The y-intercept is a critical aspect of graphing linear equations. It is defined as the point where the line crosses the y-axis. In the equation \( y = -\frac{3}{4}x - 1 \), the y-intercept is \( b = -1 \).
This point is represented on the graph as (0, -1). Every linear equation will have a y-intercept, making it a consistent starting point for graphing.
Consider the y-intercept as a foundation—it helps in plotting the entire line. Once you have located the y-intercept on the graph, use the slope to define the trajectory of the line. This systematic approach helps avoid errors in drawing the graph and ensures your line is accurately representing the equation.