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Chapter 2: Problem 18

Graph. $$y-4=-5(x-1)$$

Short Answer

Expert verified
Convert the equation to \( y = -5x + 9 \) and graph it using y-intercept 9 and slope -5.

Step by step solution

01

- Identify the Equation Form

The given equation is in point-slope form: \[ y - y_1 = m(x - x_1) \]. Here, \( m = -5 \), \( x_1 = 1 \), and \( y_1 = 4 \).
02

- Convert to Slope-Intercept Form

Convert the equation to slope-intercept form (\( y = mx + b \)) by solving for \( y \). Starting with the original equation: \[ y - 4 = -5(x - 1) \] Distribute \( -5 \): \[ y - 4 = -5x + 5 \] Add 4 to both sides: \[ y = -5x + 9 \]
03

- Identify the Slope and Y-Intercept

From the equation \( y = -5x + 9 \), identify the slope (\( m \)) and y-intercept (\( b \)). The slope \( m \) is \( -5 \), and the y-intercept \( b \) is \( 9 \).
04

- Plot the Y-Intercept

On graph paper, plot the y-intercept \( (0, 9) \). This is the point where the line crosses the y-axis.
05

- Use the Slope to Plot Another Point

Use the slope \( m = -5 \) to find another point on the line. The slope indicates a rise of \( -5 \) and a run of \( 1 \). From the y-intercept \( (0, 9) \), move down 5 units and right 1 unit to plot the point \( (1, 4) \).
06

- Draw the Line

Draw a straight line through the two points: \( (0, 9) \) and \( (1, 4) \). Extend the line across the graph.

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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 useful way to write the equation of a line when you know its slope and a point on the line. It is expressed as \[ y - y_1 = m(x - x_1) \]. Here, \((x_1, y_1)\) is a specific point on the line, and \(m\) is the line's slope.
For example, in the given equation \[ y - 4 = -5(x - 1) \], \((x_1, y_1) = (1, 4)\) and \(m = -5\). This tells us that the line passes through the point (1, 4) and has a slope of -5.
Point-slope form is particularly helpful when you are not given the y-intercept directly but have a point on the line and the slope.
Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most common ways to represent a line. It鈥檚 written as \[ y = mx + b \], where \(m\) is the slope and \(b\) is the y-intercept.
In the previous solution, we converted the point-slope form to slope-intercept form by solving for \(y\). Starting from the point-slope form \[ y - 4 = -5(x - 1) \], distribute -5 and rearrange:
\[ y - 4 = -5x + 5 \]
\[ y = -5x + 9 \]
This tells us that the slope \(m\) of the line is -5, and the y-intercept \(b\) is 9. The slope-intercept form makes it easy to graph a line by starting at the y-intercept and using the slope to find other points on the line.
Plotting Points
When graphing a linear equation, begin by plotting points on a coordinate plane. Start with the y-intercept, which is the point where the line crosses the y-axis.
In our slope-intercept form equation \[ y = -5x + 9 \], the y-intercept is 9. Plot this point at \( (0, 9)\).
Next, use the slope to find another point on the line. The slope \(m = -5\) indicates a rise of -5 and a run of 1. From \( (0, 9)\), move down 5 units and right 1 unit to plot the next point \((1, 4)\).
After plotting these points, simply draw a straight line through them to extend the line across the graph. This visual representation helps in understanding the behavior of the linear equation.

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