Problem 39 - Elementary Algebra

Question

Determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. (Objective 2) $$-y=-\frac{3}{4} x+4$$

Step-by-Step Solution

Answer

Slope: \(\frac{3}{4}\), Y-intercept: -4.

Step 1: Simplify the Equation

The given equation is \[-y = -\frac{3}{4}x + 4\]First, solve for \(y\) by multiplying every term by -1 to eliminate the negative sign in front of \(y\):\[y = \frac{3}{4}x - 4\]

Step 2: Identify the Slope and Y-intercept

The equation \(y = \frac{3}{4}x - 4\) is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. Here, the slope \(m\) is \(\frac{3}{4}\), and the y-intercept \(b\) is \(-4\).

Step 3: Plot the Y-intercept

To graph the line, start by plotting the y-intercept on the y-axis. The y-intercept is -4, so place a point at (0, -4).

Step 4: Use the Slope to Find Another Point

Since the slope is \(\frac{3}{4}\), from the point (0, -4), move up 3 units and to the right 4 units to find another point. This new point will be (4, -1).

Step 5: Draw the Line

Connect the two points (0, -4) and (4, -1) with a straight line. This is the graph of the equation \(y = \frac{3}{4}x - 4\).

Key Concepts

Graphing Linear Equations

When graphing linear equations, one of the most effective methods is to use the slope-intercept form of a linear equation, which is \(y = mx + b\). This form allows for straightforward plotting because it directly provides the slope \(m\) and the y-intercept \(b\).
To graph a linear equation, follow these steps:

• Start by identifying the y-intercept \(b\) from the equation, which is the point where the line crosses the y-axis.
• Next, use the slope \(m\) to determine the direction and steepness of the line. The slope tells you how many units to move up or down (rise) for every unit you move right (run).
• Finally, connect these points with a straight line to display the full linear relationship.

By applying this method, you can efficiently visualize any linear equation on a graph.

Y-Intercept

The y-intercept of a graph is a key feature in understanding linear equations. It is the point where the line crosses the y-axis, occurring when \(x = 0\).
In the slope-intercept form, \(y = mx + b\), the y-intercept is represented by the constant \(b\). For example, if we consider the equation \(y = \frac{3}{4}x - 4\), the y-intercept is -4. This means the line will cross the y-axis at the point (0, -4).

• To identify the y-intercept directly from the equation, look for the constant \(b\) in \(y = mx + b\).
• Graphically, place a starting point at the y-intercept value on the y-axis.

The y-intercept helps to anchor the line on the graph and serves as a starting point to use the slope to find additional points.

Slope Calculation

Calculating the slope is essential for graphing and understanding the behavior of linear equations. The slope \(m\) is the ratio of the change in \(y\) (the rise) to the change in \(x\) (the run). In the formula \(y = mx + b\), \(m\) represents this slope.
To determine the slope using the equation \(y = \frac{3}{4}x - 4\):

• Identify the coefficient of \(x\), which is the slope \(m\). Here, \(m = \frac{3}{4}\).
• This indicates that for every 4 units moved to the right, the line rises by 3 units.
• The slope is positive, suggesting that the line inclines upwards from left to right.

Understanding and calculating the slope allows you to predict how the line will extend and ensures accuracy when connecting points on the graph.