Question

Write the equation of each straight line in slope-intercept form, and make a graph. Slope \(=2.3 ; y\) intercept \(=-1.5\)

Short answer

The equation of the line in slope-intercept form is y = 2.3x - 1.5.

Step-by-step solution

Understand Slope-Intercept Form

The slope-intercept form of a straight line is given by the equation y = mx + b, where m is the slope and b is the y-intercept. In this exercise, we are given the slope (m = 2.3) and the y-intercept (b = -1.5).

Plug in the Slope and Y-Intercept

We substitute the given values into the slope-intercept form. Here, m = 2.3 and b = -1.5, so our equation becomes y = 2.3x - 1.5.

Draw the Graph

To graph the equation y = 2.3x - 1.5, start by plotting the y-intercept at (0, -1.5) on the y-axis. Then, use the slope to determine another point. With a slope of 2.3, you can rise 2.3 units on the y-axis for each unit you move to the right on the x-axis. Plot a second point using this slope and draw the line through both points.

Key concepts

Graphing Linear Equations

Graphing linear equations is an essential skill in algebra that visually represents the solution of the equation as a straight line on a coordinate plane. The process begins by identifying two important components of the line: its slope and y-intercept. In our exercise, the linear equation is already in the slope-intercept form, which is written as \( y = mx + b \), making it straightforward to graph.

To graph the equation \( y = 2.3x - 1.5 \), you start at the y-intercept, which is where the line crosses the y-axis. Here the line intersects the y-axis at \( y = -1.5 \). Plot this point on the graph. The next step is to apply the slope, \( m = 2.3 \). This slope tells you to move up 2.3 units vertically for every 1 unit you move horizontally to the right. From the y-intercept, you find another point on the line using this rise-over-run method and plot it. Once you have at least two points, you can draw the line through them, extending it across the grid to represent all solutions to the equation.

The Y-Intercept

The y-intercept is a key element in the graphing process. It specifically refers to the value of \( y \) where the line crosses the y-axis. This is where \( x = 0 \). In our example, the y-intercept is \( b = -1.5 \).

Locating the y-intercept on a graph sets an initial point from which the slope, or the steepness of the line, is applied to determine another point. Understanding the y-intercept offers insight into the initial value of the dependent variable, \( y \), before any change in the independent variable, \( x \), occurs. This is particularly useful in real-world scenarios where you're analyzing relationships between two variables, such as distance versus time or cost versus quantity.

Slope of a Line

The slope of a line is a measure of its steepness or incline and is represented by the variable \( m \) in the slope-intercept form equation. It indicates how much \( y \) changes for a unit change in \( x \). In simpler terms, it tells you how many units up or down you go (rise) for every unit you go to the right (run).

For instance, a positive slope, like \( 2.3 \) in our exercise, shows that the line rises as it moves from left to right. Conversely, a negative slope means that the line falls. The slope is crucial when graphing a linear equation since it defines the angle of the line and helps in plotting the second point after the y-intercept. Higher absolute values of slope indicate a steeper line. Understanding the concept of slope aids in identifying how rapidly variables are changing in relation to one another.