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Chapter 5: Problem 35

Graph the line containing the given point and with the given slope. $$(1,5) ; m=-3$$

Short Answer

Expert verified
The equation of the line containing the given point (1,5) and with the given slope of -3 is \(y = -3x + 8\). To graph the line, plot its y-intercept at (0,8) and follow the slope of -3 by going down 3 units and right 1 unit to plot additional points, then connect the points to form a straight line.

Step by step solution

01

Determine the equation of the line

To find the equation of the line containing the point (1,5) and with a slope of -3, we can use the point-slope form of a linear equation: \(y - y_1 = m (x - x_1)\), where (x_1, y_1) represents the given point and m represents the slope. Using the given information, we plug in the values as follows: \(y - 5 = -3 (x - 1)\)
02

Change the equation to slope-intercept form

Next, we want to convert the equation to slope-intercept form, which is given by \(\displaystyle y\ =\ mx\ +\ b\). To do this, distribute the slope and then isolate the y term: \(y - 5 = -3 (x - 1)\) \(y - 5 = -3x + 3\) Now, add 5 to both sides of the equation to isolate y: \(y = -3x + 8\) Now, the equation of the line is in slope-intercept form with a slope (m) of -3 and a y-intercept (b) of 8.
03

Graph the line

To graph the line, follow these steps: 1. Plot the y-intercept (b) on the y-axis. In this case, the y-intercept is 8, so place a point at (0,8). 2. Use the slope (m) to plot additional points on the line. Since the slope is -3, this means that, starting from the y-intercept, the line goes down 3 units for every 1 unit it moves to the right. 3. Connect the points on the graph to form a straight line. By following these steps, you will have successfully graphed the line containing the given point (1,5) and with the given slope of -3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-Slope Form
The point-slope form is a convenient way to find the equation of a line when you know the slope and a single point on the line. The general formula is \(y - y_1 = m(x - x_1)\), where:
  • \(x_1, y_1\) is the known point on the line.
  • \(m\) is the slope.
Simply plug in the known values to create an equation. For example, if the point is (1, 5) and the slope, \(m\), is -3, the equation becomes \(y - 5 = -3(x - 1)\).
This format directly ties the change in \(y\) to the change in \(x\), which makes it very intuitive for understanding how the slope affects the line.
Slope-Intercept Form
The slope-intercept form, \(y = mx + b\), is probably the most popular way to express the equation of a line. In this form:
  • \(m\) represents the slope.
  • \(b\) is the y-intercept, where the line crosses the y-axis.
To convert from point-slope to slope-intercept form, simply solve for \(y\) by distributing and rearranging the terms.
For instance, from the previous example, the equation \(y - 5 = -3(x - 1)\) becomes \(y = -3x + 8\) after simplifying.
This format is fantastic for graphing and quick calculations, as it clearly shows both the slope and the exact point where the line intercepts the y-axis.
Graphing Lines
Graphing lines is a practical application of understanding linear equations. The process usually involves a few simple steps:
  • Start by plotting the y-intercept \(b\) on the y-axis.
  • Use the slope \(m\) to determine how the line moves across the graph.
For a slope \(m = -3\), it moves 3 units down for every 1 unit it moves to the right. Connect these points to draw the line.
Always begin with the y-intercept, which provides a stable starting point, and then use the slope to mark another point for accuracy.
Once the line is drawn, verify that it indeed intersects the y-axis at \(b\) and correctly reflects the given slope. This ensures both accuracy and a deeper understanding of how line equations translate to graphs.

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