Question

Write the equation of each straight line in slope-intercept form, and make a graph. Slope \(=3 ; y\) intercept \(=-1\)

Short answer

The equation of the line in slope-intercept form is y = 3x - 1.

Step-by-step solution

Identify the Slope and y-intercept

From the given information, we know the slope (m) is 3 and the y-intercept (b) is -1. The Slope-Intercept form of a line is given by the equation: y = mx + b.

Plug in the Values into the Slope-Intercept Form

By plugging the values of m and b into the general equation y = mx + b, we get the specific equation for our line: y = 3x - 1.

Plot the y-intercept on the Graph

First, we plot the y-intercept point (0, -1) on the graph. This will be the point where the line crosses the y-axis.

Use the Slope to Determine Another Point

Using the slope of 3, which means the rise over run is 3, from the y-intercept move 3 units up for the rise and 1 unit to the right for the run to get the next point (1, 2).

Draw the Line

Draw a straight line through the points (0, -1) and (1, 2). This is the graph of the line with the equation y = 3x - 1.

Key concepts

Linear Equations

Linear equations are the foundation of algebra and represent relationships with a constant rate of change. They can be recognized by an equation in which the highest power of the variable is one, forming a straight line when graphed on a coordinate plane. The standard form of a linear equation is Ax + By = C , where A, B, and C are constants, and x and y are variables.

When working with linear equations, it is often convenient to express them in slope-intercept form, which is y = mx + b. This form allows for quick identification of the slope, or steepness, of the line, as well as the y-intercept, where the line crosses the y-axis. For students, this form simplifies the process of graphing, as shown in the exercise where a linear equation with a given slope and y-intercept is transformed directly into a graphable line. Understanding the components of the slope-intercept form is essential for analyzing linear relationships and is a crucial skill in mathematics.

Graphing Lines

Graphing lines is a visual way of representing linear equations. It's like drawing a mathematical map that shows you where the equation 'lives' on a two-dimensional plane. To graph a line, you typically need two points. After plotting these points, you draw a straight line through them, which extends infinitely in both directions.

In relation to the slope-intercept form, graphing becomes a straightforward task. Start by plotting the y-intercept (the point where the line crosses the y-axis). From there, you'll use the slope to determine the next point. If the slope is positive, the line rises as it moves from left to right; if negative, it falls. For instance, with a slope of 3, you'd move up 3 units (the rise) and to the right 1 unit (the run), which is how we found the point (1, 2) from the y-intercept (0, -1) in the example. This point-to-point approach simplifies creating the visual representation of a linear equation.

Slope and y-intercept

The slope and y-intercept are crucial in defining the characteristics of a line, and they make up the slope-intercept form of a linear equation, y = mx + b. The slope, represented by the variable m, measures how steep the line is. In the context of a graph, it is calculated as the 'rise' over the 'run', or how much the line goes up or down for a given horizontal distance. Positive slopes rise upwards to the right, while negative slopes fall downwards to the right.

The y-intercept, represented by b, indicates where a line crosses the y-axis. This is where the value of x is zero. When graphing, it's the starting point used to draw the line. For example, a y-intercept of -1 means that the line crosses the y-axis at (0, -1). By combining both the slope and y-intercept, students can quickly and accurately draw the graph of a linear equation, as well as interpret the equation's relationship by understanding the rate of change and the starting point.