Problem 74 - Algebra 2

Question

Write an equation for each line. Then graph the line. \(m=-\frac{3}{2},\) through \((0,-1)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -\frac{3}{2}x - 1\). For graphing, start at the point \((0, -1)\) and follow the slope of \(-\frac{3}{2}\) to draw the line.

Step 1: Write the slope-intercept form of the equation

Start with the slope-intercept form of the linear equation, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Use the given slope

Substitute the given slope \(m = -\frac{3}{2}\) into the equation from Step 1, resulting in \(y = -\frac{3}{2}x + b\).

Step 3: Substitute the given point to find the y-intercept

Plug in the coordinates of the given point \((0, -1)\) into the equation to find the y-intercept \(b\). Since \(x=0\), the equation is \(-1 = -\frac{3}{2}\cdot0 + b\). Solve for \(b\) to find that \(b = -1\).

Step 4: Write the final equation

Now that you have the slope and y-intercept, write the final equation of the line as \(y = -\frac{3}{2}x - 1\).

Step 5: Graph the line

To graph the line, start at the y-intercept \((0, -1)\). Then, from this point, use the slope to determine the next point. Since the slope is \(-\frac{3}{2}\), move down 3 units and to the right 2 units to plot another point. Draw the line through these points to complete the graph.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that represent a line in a coordinate space. They illustrate a relationship between two variables, typically x and y, and can be depicted in a variety of forms, including slope-intercept form, point-slope form, and the standard form.

One of the simplest and most commonly used forms is the slope-intercept form, which is represented as \( y = mx + b \). In this equation, \( m \) symbolizes the slope of the line, and \( b \) denotes the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful because it provides an immediate visualization of the line's slope and y-intercept, making it easy to graph.

Graphing Lines

Graphing a line on a coordinate plane involves plotting points that satisfy the equation of the line and then connecting these points in a straight path. When employing the slope-intercept form \( y = mx + b \), the process begins with marking the y-intercept, the point \( (0, b) \), on the y-axis.

Step by Step Graphing

• Locate the y-intercept \( (0, b) \) on the y-axis and plot the point.
• Utilize the slope \( m \), which is the rate of change along the line, to identify another point. Move vertically by the numerator and horizontally by the denominator of the slope from the y-intercept.
• Conjoin the points with a ruler to form a straight line extending in both directions.

Through this method, one can accurately sketch the line that correlates to the linear equation.

Slope of a Line

The slope of a line is a measure of its steepness or incline, represented by the letter \( m \) in the slope-intercept form of a linear equation. Mathematically, slope is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.

For any two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) is determined by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). If the slope is positive, this indicates that the line ascends from left to right, while a negative slope indicates that the line descends. A slope of zero implies a horizontal line, and undefined slope signifies a vertical line.

Y-Intercept

The y-intercept of a line is the specific point where the line crosses the y-axis on a graph. Mathematically, it is the value of \( y \) in the linear equation when \( x \) is zero. The y-intercept is a critical aspect of the slope-intercept form of a linear equation, embodied as \( b \) in the equation \( y = mx + b \).

Identifying the y-intercept is straightforward when you have an equation or two points. If provided a point and a slope, you can insert the point's coordinates into the slope-intercept form to solve for \( b \), thus determining the y-intercept. This key feature establishes the starting point for graphing a line and facilitates a visual understanding of how the line is positioned relative to the y-axis.