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Chapter 2: Problem 28

Graph the line that contains the point P and has slope \(\mathrm{m}\). $$ P=(1,3) ; m=-\frac{2}{5} $$

Short Answer

Expert verified
Use the point-slope form to get \( y = -\frac{2}{5}x + 3.4 \), then plot the points (0, 3.4) and (5, 1.4).

Step by step solution

01

- Write the Point-Slope Form Equation

The point-slope form of a line's equation is given by \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point on the line and \( m \) is the slope. Here, the point \( P = (1, 3) \) and the slope \( m = -\frac{2}{5} \). Substitute these values into the equation: \[ y - 3 = -\frac{2}{5}(x - 1) \]
02

- Rearrange the Equation into Slope-Intercept Form

To make it easier to plot the line, rearrange the equation into slope-intercept form: \[ y = mx + b \]. Start by distributing the slope \( m \) to \( (x - 1) \): \[ y - 3 = -\frac{2}{5}x + \frac{2}{5} \]. Then, isolate \( y \) by adding 3 to both sides: \[ y = -\frac{2}{5}x + \frac{2}{5} + 3 \]. Combine the constants (\( \frac{2}{5} + 3 = 3.4 \)): \[ y = -\frac{2}{5}x + 3.4 \]
03

- Identify Key Points to Plot

Use the slope-intercept form \( y = -\frac{2}{5}x + 3.4 \) to identify key points. Start with the y-intercept, which is 3.4. When \( x=0 \), \( y = 3.4 \). Next, use the slope \( -\frac{2}{5} \) to find another point: if you move 5 units to the right (positive direction on the x-axis), move down 2 units (negative direction on the y-axis). From the y-intercept (0, 3.4), a point is (5, 1.4).
04

- Plot the Points and Draw the Line

Plot the points (0, 3.4) and (5, 1.4) on the graph. Since a line is straight, you only need these two points to draw the line. Connect the points with a straight line, extending in both directions, and you have the graph of the line.

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Key Concepts

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

point-slope form
Understanding the point-slope form of a linear equation is crucial for graphing lines efficiently. The point-slope form is given by the equation: ewline ewline ewline ewline ewline ewline ewline
slope-intercept form
The next important concept to understand is the slope-intercept form of a linear equation. Slope-intercept form is faster to graph because it conveniently shows the y-intercept and slope. The equation is:ewline ewline ewline
plotting points
Finally, plotting points accurately on a graph ensures your representation of the line is correct. When you start plotting, follow these steps: ewline ewline This will help your graph be accurate and neat.

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