Problem 46 - Elementary Algebra

Question

Graph the line that passes through the given point and has the given slope. (Objective 3 ) $$(-3,4), m=-\frac{3}{2}$$

Step-by-Step Solution

Answer

Graph the line using the point-slope form equation, then plot the line based on the slope.

Step 1: Understand the problem

We need to graph the line that passes through the point (-3, 4) and has a slope of \( m = -\frac{3}{2} \). The slope-intercept form of a line \( y = mx + b \) will be useful to understand its graph.

Step 2: Use point-slope form

To begin sketching the line, use the point-slope form of a line equation: \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1) \) is the point (-3, 4), and the slope, \( m = -\frac{3}{2} \).

Step 3: Substitute given values

Substitute \((x_1, y_1) = (-3, 4) \) and \( m = -\frac{3}{2} \) into the equation: \[ y - 4 = -\frac{3}{2}(x + 3) \]

Step 4: Simplify the equation

Expand and simplify the equation:\[ y - 4 = -\frac{3}{2}x - \frac{9}{2} \]Add 4 to both sides to isolate \( y \):\[ y = -\frac{3}{2}x - \frac{9}{2} + 4 \] To combine, convert 4 to a fraction:\[ y = -\frac{3}{2}x - \frac{9}{2} + \frac{8}{2} \] So: \[ y = -\frac{3}{2}x - \frac{1}{2} \]

Step 5: Graph the equation

Start by plotting the known point (-3, 4) on the coordinate plane. Using the slope \( m = -\frac{3}{2} \), go down 3 units and right 2 units from the point to plot another point. Draw a line through these points to represent the equation\( y = -\frac{3}{2}x - \frac{1}{2} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a particular way to express the equation of a line. The general formula is given by\[ y = mx + b \]where:

• \( y \) represents the dependent variable.
• \( m \) stands for the slope of the line.
• \( x \) symbolizes the independent variable.
• \( b \) designates the y-intercept of the line, which is the point where the line crosses the y-axis.

The beauty of the slope-intercept form is that it quickly reveals two critical characteristics of a line: its steepness (slope) and its y-intercept.
To find the slope-intercept form, begin with the equation in another form, like the point-slope form, and rearrange it to solve for \( y \). This makes it easy to graph the line, as you can start by plotting the y-intercept and then simply use the slope to find additional points.

Point-Slope Form

The point-slope form is a powerful tool for writing the equation of a line when you know a point on the line and the slope. It is written as:\[ y - y_1 = m(x - x_1) \]Here:

• \( (x_1, y_1) \) are the coordinates of the known point on the line.
• \( m \) is the slope of the line.

This form is particularly useful for drawing a line when you have a point and a slope without needing to first find the y-intercept.
In our problem, given the point (-3, 4) and the slope \( m = -\frac{3}{2} \), you can substitute these values directly into the equation like this:\[ y - 4 = -\frac{3}{2}(x + 3) \]You can then convert this to the slope-intercept form by isolating \( y \). This gives us a clear visual representation and makes the graphing process straightforward.

Plotting Points

Plotting points is a fundamental aspect of graphing linear equations. It involves locating points on the coordinate plane that satisfy the equation of the line.To graph a line, you start by plotting a known point, often derived from the line equation itself, such as the y-intercept or another given point.

• The coordinates are given in the form \((x, y)\), showing the position in terms of horizontal and vertical distances from the origin (0,0).
• By using the slope, you can determine the direction and steepness of the line. For the slope \(-\frac{3}{2}\), it indicates that for every 2 units you move to the right on the x-axis, you move 3 units down on the y-axis.

Once a few points are plotted using the slope, you connect them to draw the line that represents the equation.
This sequence of plotting initial points followed by additional points according to the slope is key to graphing any line accurately. It provides a visual way to confirm your calculations and understand the orientation of the line.