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Chapter 11: Problem 265

In the following exercises, graph the line given a point and the slope. $$ (1,5) ; m=-3 $$

Short Answer

Expert verified
Plot the point (1, 5), use the slope to find another point (2, 2), and draw the line through these points.

Step by step solution

01

Understand the Point-Slope Form

The point-slope form of a linear equation is given by y - y_1 = m(x - x_1), where y_1 is the y-coordinate, x_1 is the x-coordinate of the given point, and m is the slope.
02

Substitute the Given Values

Substitute the given point (x_1, y_1) = (1, 5) and the given slope m = -3 into the point-slope form: y - 5 = -3(x - 1).
03

Simplify the Equation

Simplify the equation to the slope-intercept form y = mx + b. Starting with y - 5 = -3(x - 1), distribute -3: y - 5 = -3x + 3. Add 5 to both sides: y = -3x + 8. This is the equation of the line.
04

Plot the Point

On a graph, plot the given point (1, 5).
05

Use the Slope

The slope m = -3 means that for every 1 unit you move to the right on the x-axis, you move 3 units down on the y-axis. From the point (1, 5), move 1 unit to the right to (2, 5) and then 3 units down to (2, 2). Plot this point as well.
06

Draw the Line

Draw a line through the points (1, 5) and (2, 2). This is the graph of the line with the equation y = -3x + 8.

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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 of a linear equation is a way to write the equation of a line when you know the slope and a point on the line. This form is expressed as:
\( y - y_1 = m(x - x_1) \)
Here:
  • \( y_1 \) is the y-coordinate of the given point
  • \( x_1 \) is the x-coordinate of the given point
  • \( m \) is the slope of the line
For example, if we have the point (1, 5) and a slope of -3, we substitute these values into the point-slope formula:
\( y - 5 = -3(x - 1) \).
In this equation, \( x_1 = 1 \), \( y_1 = 5 \), and \( m = -3 \). This form allows you to quickly write the equation of the line without needing to graph it first.
Slope-Intercept Form
Once you have the point-slope form of an equation, it's often useful to convert it into the slope-intercept form, especially for graphing purposes. The slope-intercept form is:
\( y = mx + b \)
Here:
  • \( m \) represents the slope of the line
  • \( b \) represents the y-intercept, the point where the line crosses the y-axis
To convert from point-slope form to slope-intercept form, you simplify the equation:
Starting with \( y - 5 = -3(x - 1) \), distribute -3:
\( y - 5 = -3x + 3 \)
Add 5 to both sides to isolate y:
\( y = -3x + 8 \)
Now, the equation \( y = -3x + 8 \) is in slope-intercept form, making it easier to graph.
Coordinate Plotting
Graphing a linear equation involves plotting points and drawing a line through them. Here鈥檚 how to do it:
  • First, plot the given point, in this case, (1, 5). Place a dot at this coordinate on your graph.
  • Next, use the slope to find another point. The slope -3 means that for every 1 unit you move to the right on the x-axis, you move 3 units down on the y-axis.
  • From the point (1, 5), move 1 unit to the right to (2, 5) and then 3 units down to (2, 2). Plot this new point.
  • Draw a line through the points (1, 5) and (2, 2). This line represents the equation \( y = -3x + 8 \).
By plotting points and using the slope, you can accurately draw the line represented by any linear equation.

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Most popular questions from this chapter

In the following exercises, graph each equation. $$ y=5 $$

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