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Magazine Chapter 3: Problem 96
In the following exercises, graph each line with the given point and slope. $$ (-3,-5) ; m=\frac{3}{2} $$
Short Answer
Expert verified
The line equation is \( y = \frac{3}{2} x - \frac{1}{2} \).
Step by step solution
01
Understand the Line Equation
To graph a line given a point and a slope, you'll start by understanding the slope-intercept form of the equation of the line, which is given by \[ y = mx + b \]. Here, \( m \) is the slope and \( b \) is the y-intercept.
02
Substitute the Slope
Substitute the given slope \( m = \frac{3}{2} \) into the slope-intercept form. The equation now looks like \[ y = \frac{3}{2}x + b \].
03
Substitute the Point into the Equation
Now use the given point \( (-3, -5) \) to find the y-intercept \( b \). Substitute \( x = -3 \) and \( y = -5 \) into the equation: \[ -5 = \frac{3}{2}(-3) + b \].
04
Solve for y-intercept
Solve the equation for \( b \): 1. \[ -5 = \frac{3}{2}(-3) + b \] 2. Calculate \( \frac{3}{2}(-3) \): \[ -5 = -\frac{9}{2} + b \] 3. Add \( \frac{9}{2} \) to both sides to isolate \( b \): \[ -5 + \frac{9}{2} = b \] 4. Convert \( -5 \) to a fraction with the same denominator: \[ -\frac{10}{2} + \frac{9}{2} = b \] 5. Solve: \[ b = -\frac{1}{2} \].
05
Write the Final Equation
Substitute \( b = -\frac{1}{2} \) back into the slope-intercept form to get the final equation of the line: \[ y = \frac{3}{2} x - \frac{1}{2} \].
06
Graph the Line
To graph the line, start by plotting the y-intercept \( (0, -\frac{1}{2}) \). From there, use the slope \( \frac{3}{2} \) to find another point. The slope means that for every 2 units you move to the right, you move up 3 units. So from \( (0, -\frac{1}{2}) \), move right 2 units to \( (2, -\frac{1}{2}) \) and up 3 units to \( (2, \frac{5}{2}) \). Plot this second point and draw a line through the two points.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most common ways to express the equation of a line. The general formula is given by \( y = mx + b \) where \( m \) represents the slope and \( b \) represents the y-intercept. Understanding this form is crucial because it allows you to quickly determine the rate at which y values change as x values change (which is your slope) and where the line crosses the y-axis (which is your y-intercept). When given a slope and a point, inserting these values into the formula helps you determine the full equation of the line, making graphing much simpler.
Y-Intercept
The y-intercept is where the line crosses the y-axis. This point occurs when \( x = 0 \). In the equation \( y = mx + b \), \( b \) is the y-intercept. Identifying the y-intercept is a key step in graphing a line because it provides a starting point for the line on the graph.
For example, in the final equation \( y = \frac{3}{2} x - \frac{1}{2} \), the y-intercept is \( -\frac{1}{2} \). This means the line crosses the y-axis at the point \( (0, -\frac{1}{2}) \).
For example, in the final equation \( y = \frac{3}{2} x - \frac{1}{2} \), the y-intercept is \( -\frac{1}{2} \). This means the line crosses the y-axis at the point \( (0, -\frac{1}{2}) \).
Coordinate Points
Coordinate points are specific pairs of x and y values that lie on a line. In the coordinate plane, any line can be represented by an infinite number of points, but you only need a few to graph it.
For example, to graph the line with slope \( \frac{3}{2} \) and a point \( (-3, -5) \), you first find the y-intercept and then use the slope to find more points:
For example, to graph the line with slope \( \frac{3}{2} \) and a point \( (-3, -5) \), you first find the y-intercept and then use the slope to find more points:
- Starting from the y-intercept \( (0, -\frac{1}{2}) \)
- Move right 2 units to \( (2, -\frac{1}{2}) \)
- Then move up 3 units to \( (2, \frac{5}{2}) \). This gives another point on the line
Solving for b
When given a slope and a point, to find the y-intercept \(b\), you substitute the slope and the given point into the slope-intercept form \( y = mx + b \):
- Using the point \( (-3, -5) \)
- And the slope \( \frac{3}{2} \)
- Substitute \( x = -3 \) and \( y = -5 \) into \( y = \frac{3}{2}x + b \): \( -5 = \frac{3}{2}(-3) + b \)
- Calculate \( \frac{3}{2}(-3) = -\frac{9}{2} \)
- Add \( \frac{9}{2} \) to both sides: \( -5 + \frac{9}{2} = b \)
- Convert \( -5 \) to a fraction: \( -\frac{10}{2} \)
- Combine: \( -\frac{10}{2} + \frac{9}{2} = -\frac{1}{2} \)
- Thus, \( b = -\frac{1}{2} \)
