Problem 14 - Technical Mathematics with Calculus

Question

Write the equation of each straight line and make a graph. Slope \(=-1 ; y\) intercept \(=2\)

Step-by-Step Solution

Answer

The equation of the line is y = -x + 2. To graph it, plot the y-intercept (0, 2) and then use the slope to find another point such as (1, 1), and draw the line through these points.

Step 1: Understand the Slope-Intercept Form

The slope-intercept form is written as y = mx + b, where m is the slope and b is the y-intercept. Our given slope (m) is -1 and the y-intercept (b) is 2.

Step 2: Substitute the Slope and Y-Intercept Into the Formula

Substitute m = -1 and b = 2 into the slope-intercept form of a line equation y = mx + b to obtain the equation of the line.

Step 3: Write the Equation of the Line

After substituting, we get the equation of the line as y = -1x + 2, which can simply be written as y = -x + 2.

Step 4: Plot the Y-Intercept on the Graph

Begin by plotting the y-intercept, which is the point (0, 2) on the y-axis.

Step 5: Use the Slope to Find Another Point

Starting from the y-intercept (0, 2), use the slope -1 (which means down 1 unit and right 1 unit) to find another point on the line. This leads to the point (1, 1). Plot this point on the graph.

Step 6: Draw the Line

Draw a straight line through the two points (0, 2) and (1, 1). This line represents the graph of the equation y = -x + 2.

Key Concepts

Graphing Linear Equations

Understanding how to graph linear equations is a fundamental skill in algebra. A linear equation can be graphed by finding two or more points that satisfy the equation and then connecting these points with a straight line. The most common form used for graphing is the slope-intercept form, expressed as y = mx + b, where m is the slope, and b is the y-intercept. When graphing, one starts by identifying the y-intercept, which is where the line crosses the y-axis, and from there, uses the slope to determine another point.

For instance, with a slope of -1 and a y-intercept of 2, we would plot the point (0, 2) on the y-axis. To find the next point, we follow the slope, moving down 1 unit and right 1 unit, landing at the point (1, 1). By plotting these points and drawing a line through them, the linear equation is graphically represented on a coordinate plane.

Slope of a Line

The slope of a line is a measure of its steepness and direction. Algebraically, it is expressed as m in the slope-intercept form equation y = mx + b. The slope is calculated as the ratio of the rise (vertical change) to the run (horizontal change) between any two points on a line. A positive slope means the line inclines upwards as it moves from left to right. Conversely, a negative slope indicates the line declines.

A slope of -1 means that for every unit the line moves horizontally to the right, it moves down one unit vertically, creating a line that declines diagonally. Graphically, this can be seen as the line descending in the standard coordinate plane, with every rightward step matched by an equivalent downward step.

Y-intercept

The y-intercept is a key component in both the slope-intercept form of a linear equation and in the graphing process. It represents the value of y at which the line crosses the y-axis when x is zero. Notated as b in the equation y = mx + b, it provides a starting point for graphing and conceptualizing the line's position relative to the origin.

Take, for example, a y-intercept of 2. This intercept tells us that the line will cross the y-axis at the point where y is 2, namely at (0, 2). This is the first plotted point when drawing a line on a graph, from which we then apply the slope to determine the line's direction and additional points.

Plotting Points on a Graph

Plotting points is the action of marking specific locations on a coordinate plane based on their x and y values. In the context of linear equations, plotting points is useful for visualizing the line the equation represents. To plot points accurately, one should locate the exact position on the x-axis (horizontal) for the x-coordinate and the same on the y-axis (vertical) for the y-coordinate, then mark where these two positions intersect.

For a line with the equation y = -x + 2, after plotting the y-intercept at (0, 2), another point is found using the slope. In this case, starting from (0, 2), and moving one step right (positive x-direction) and one down (negative y-direction), we reach (1, 1), which we then plot. Connecting the dots creates a visual representation of the equation on the graph.