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Chapter 11: Problem 259

In the following exercises, graph the line given a point and the slope. $$ (0,3) ; m=-\frac{2}{5} $$

Short Answer

Expert verified
The equation of the line is \( y = -\frac{2}{5}x + 3 \).

Step by step solution

01

Understand the Point-Slope Form

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

Identify the Given Values

From the given exercise, the point is \((0,3)\) and the slope \( m = -\frac{2}{5} \). Therefore, \( x_1 = 0 \) and \( y_1 = 3 \).
03

Plug in the Values into the Point-Slope Form

Substitute the point \((0, 3)\) and the slope \( -\frac{2}{5} \) into the point-slope form: \[ y - 3 = -\frac{2}{5}(x - 0) \]
04

Simplify the Equation to Slope-Intercept Form

Simplify the equation to get it into the slope-intercept form \( y = mx + b \): \[ y - 3 = -\frac{2}{5}x \] \[ y = -\frac{2}{5}x + 3 \]
05

Graph the Line

Use the slope-intercept form \( y = -\frac{2}{5}x + 3 \) to graph the line: 1. Start at the y-intercept (0, 3).2. Using the slope \( -\frac{2}{5} \), from (0, 3), move down 2 units and right 5 units to find another point on the graph.3. Draw the line through the points.

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Key Concepts

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

Point-Slope Form
In mathematics, the point-slope form is an incredibly useful way of expressing the equation of a line. It helps to create a linear equation when you know one point on the line and the slope. The formula is: \( y - y_1 = m(x - x_1) \), where:
  • \(x_1\) and \(y_1\) represent coordinates of a specific point on the line.
  • \(m\) stands for the slope of the line.
Let's take the given example: the point is (0,3) and the slope \(m = - \frac{2}{5}\). Plugging these values into the formula, it becomes: \( y - 3 = -\frac{2}{5}(x - 0) \). From here, simplifying further can transform this into other forms, like slope-intercept form, which can make graphing easier.
Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most straightforward methods to represent a line. The standard formula looks like this: \( y = mx + b \). Here,
  • \(m\) is the slope of the line.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
To convert from point-slope to slope-intercept, begin with the point-slope form: \( y - 3 = -\frac{2}{5}(x - 0) \). Simplify by distributing and adding 3 to isolate \(y\): \( y - 3 = -\frac{2}{5}x \) then \( y = -\frac{2}{5}x + 3 \). This shows the equation in slope-intercept form, making it easy to graph by identifying the slope and y-intercept.
Plotting Points
Plotting points on a graph is a fundamental skill to visualize linear equations. Here's a simple approach to graphing lines step-by-step:
  • Start by identifying the y-intercept, in this example, it's (0,3). Plot this point on the y-axis.
  • Next, use the slope to find additional points. A slope of \(- \frac{2}{5}\) means you move down 2 units for every 5 units you move right. From (0,3), you move to the point (5,1).
  • Plot the second point and draw a straight line through both points. This line represents the equation \( y = - \frac{2}{5}x + 3 \).
By following these steps, plotting points accurately ensures you can graph any linear equation effectively.

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Most popular questions from this chapter

In the following exercises, use the slope formula to find the slope of the line between each pair of points. $$ (3,5),(4,-1) $$

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

In the following exercises, find three solutions to each equation and then graph each line. $$ y=-3 x $$

In the following exercises, find three solutions to each linear equation. $$ y=-x-1 $$

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