Problem 17 - Precalculus with Limits

Question

In Exercises 17-28, find the slope and \(y\)-intercept (if possible) of the equation of the line. Sketch the line. \( y = 5x + 3 \)

Step-by-Step Solution

Answer

The slope of the line is 5 and the y-intercept is 3.

Step 1: Identify the Slope

The slope is the coefficient in front of \( x \) in the equation. From the given equation \( y = 5x + 3 \), the slope \( m \) is 5.

Step 2: Identify the y-intercept

The y-intercept is the value of \( y \) when \( x \) equals zero. It's represented by the standalone number in the equation. In the case of this equation \( y = 5x + 3 \), the y-intercept \( b \) is 3.

Step 3: Sketch the line

Start at the y-intercept (0,3), and use the slope to find another point. Slope is rise over run, meaning for every 5 units increase in \( y \), there is 1 unit increase in \( x \). Starting at (0,3), go up 5 units (to \( y = 8 \)) and right 1 unit (to \( x = 1 \)). So the second point is located at (1,8). Draw a line through these points to establish the graph.

Key Concepts

Linear Equations

Linear equations are foundational in algebra and represent relationships where the rate of change is constant. A linear equation in two variables, typically denoted as 'x' and 'y', can be written in the form of \( y = mx + b \), where 'm' represents the slope and 'b' signifies the y-intercept. The slope defines the steepness of the line and whether it rises or falls as one progresses along it. On the other hand, the y-intercept indicates the point where the line crosses the y-axis. Put simply, it is the value of 'y' when 'x' equals zero.

In the given exercise, the linear equation presented \( y = 5x + 3 \) shows a relation where for every unit increase in 'x', the value of 'y' increases by five times that amount. Understanding these equations allows for predicting and interpreting results within numerous real-world contexts, such as calculating speed, forecasting expenses, or analyzing trends.

Graphing Lines

When graphing lines, particularly those represented by linear equations, you begin by plotting the y-intercept, which is the point where the line crosses the y-axis. This is followed by using the slope to determine the angle and direction in which the line progresses. The slope, often described as the 'rise over run', compares the vertical change ('rise') to the horizontal change ('run').

To graph the line from the equation \( y = 5x + 3 \), you would first plot the point (0, 3), then from there move up 5 units and to the right 1 unit to find the next point (1, 8). Connecting these points will help draw the entire line on a graph, providing a visual representation that assists in understanding the behavior of the equation. Graphing is especially beneficial when comparing multiple lines, as it showcases their interactions, such as intersections or parallelism, which correspond to solutions or relationships within the equations.

Slope-Intercept Form

The slope-intercept form is particularly useful for quickly graphing a linear equation. It is expressed as \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept — the exact point where the line crosses the y-axis. This form is distinct in its clarity: it allows you to immediately identify the steepness of the line and where to begin plotting on the graph.

The given exercise with the equation \( y = 5x + 3 \) is a textbook example of the slope-intercept form. The number 5, being multiplied by 'x', denotes that the line ascends 5 units vertically for every single unit of horizontal movement, which is a relatively steep incline. On the other hand, the number 3 indicates that the line intersects the y-axis at the point (0,3). Emphasizing these two parameters empowers students to swiftly visualize and sketch the behavior of linear equations on a graph.