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Chapter 5: Problem 41

Sketch the graph of the line satisfying the given conditions. Passing through \((1,-2)\) with slope \(\frac{-2}{3}\)

Short Answer

Expert verified
The line passes through \(1, -2\) with a slope of \(\frac{-2}{3}\). Equation: \(y = \frac{-2}{3}x - \frac{4}{3}\).

Step by step solution

01

Identify given information

The given point is \(1, -2\), and the slope of the line is \(\frac{-2}{3}\text{\)}.
02

Use point-slope form

The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the given point and \m\ is the slope. Substitute the values: \(y + 2 = \frac{-2}{3}(x - 1)\).
03

Distribute and simplify

Distribute the slope on the right side: \(y + 2 = \frac{-2}{3}x + \frac{2}{3}\).
04

Isolate y

Subtract 2 from both sides to isolate \y\: \(y = \frac{-2}{3}x + \frac{2}{3} - 2\).\text{Convert } -2 \text{ to} \ \frac{-6}{3}\ \text{and simplify the y-intercept term.} \(y = \frac{-2}{3}x - \frac{4}{3}\).This is the slope-intercept form.
05

Plot the given point

Mark the point \(1, -2\) on the graph.
06

Use slope to find another point

From point \(1, -2\), use the slope \(\frac{-2}{3}\text{\)} to plot another point: move 3 units to the right \ and 2 units down to reach \(4, -4\).
07

Draw the line

Draw a straight line passing through the points \((1, -2)\) and \((4, -4)\).

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Key Concepts

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

Point-Slope Form of a Linear Equation
The point-slope form is very useful for writing an equation when you have a point and the slope of the line. It is given by the formula: \( y - y_1 = m(x - x_1) \). Here,
  • \( (x_1, y_1) \) are the coordinates of the given point
  • \( m \) is the slope of the line
You simply substitute the point and the slope into the formula to get the equation. In the provided exercise, the given point is \( (1, -2) \) and the slope is \( \frac{-2}{3} \). Substituting these values in, we get: \( y + 2 = \frac{-2}{3}(x - 1) \), which is our equation in point-slope form.
Slope-Intercept Form of a Linear Equation
The slope-intercept form is another popular way to write the equation of a line. Its formula is \( y = mx + b \), where
  • \( m \) is the slope
  • \( b \) is the y-intercept (where the line crosses the y-axis)
After obtaining the point-slope form, you can convert it to the slope-intercept form by solving for \( y \). For example:
Starting from \( y + 2 = \frac{-2}{3}(x - 1) \), distribute the \( \frac{-2}{3} \) on the right side:
\( y + 2 = \frac{-2}{3}x + \frac{2}{3} \)
Subtract 2 from both sides to isolate \( y \):
\( y = \frac{-2}{3}x + \frac{2}{3} - 2 \)
To simplify, convert \( -2 \) to a fraction with the same denominator:
\( -2 = \frac{-6}{3} \)
Simplifying further, you'll get:\( y = \frac{-2}{3}x - \frac{4}{3} \). Now the equation is in slope-intercept form.
Plotting Points on a Graph
Plotting points is a fundamental skill in graphing linear equations. Each point has coordinates \( (x, y) \). For example, the point \( (1, -2) \) means you go 1 unit to the right (along the x-axis) and 2 units down (along the y-axis). To plot this point on a graph:
  • Start from the origin \( (0, 0) \)
  • Move 1 unit to the right
  • Move 2 units down
Mark this point. In the exercise, this is our starting point. After this, we can use the slope to find more points on the line and eventually draw it.
Calculating Slope
The slope of a line tells us how steep the line is. It is calculated as the 'rise' over the 'run' between two points on the line. The formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Where:
  • \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on the line
For example, if the points are \( (1, -2) \) and \( (4, -4) \), we can use the formula to find the slope:
\( m = \frac{-4 - (-2)}{4 - 1} = \frac{-4 + 2}{4 - 1} = \frac{-2}{3} \). This tells us that for every 3 units we move to the right, we move 2 units down.
Graphing Linear Equations
Graphing linear equations involves plotting points and drawing a straight line through them. To graph a line based on its equation:
  • Convert the equation to slope-intercept form to identify the slope and y-intercept
  • Plot the y-intercept on the y-axis
  • Use the slope to find another point starting from the y-intercept
  • Draw a straight line through the points
For instance, given \( y = \frac{-2}{3}x - \frac{4}{3} \):
  • The y-intercept is \( - \frac{4}{3}\), i.e., mark the point \( (0, - \frac{4}{3}) \)
  • From this point, use the slope \( \frac{-2}{3} \) to find another point, i.e., move 3 units right and 2 units down
You can repeat this process to ensure the line is accurate. Then, connect the points with a straight line, and you've successfully graphed the linear equation!

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Most popular questions from this chapter

Plot a few points that satisfy the equation \(y=x^{3} .\) Do you think the graph of this equation is a straight line? Explain.

How could you use the idea of slope to show that the three points \((-1,-2)\) \((2,0),\) and \((5,2)\) all lie on a straight line?

An efficiency expert finds that the average number of defective items produced in a factory is approximately linearly related to the average number of hours per week the employees work. If the employees average 34 hours of work per week, then an average of 678 defective items are produced. If the employees average 45 hours of work per week, then an average of 834 defective items are produced. (Round to the nearest tenth) (a) Write an equation relating the average number of defective items \(d\) and the average number of hours \(h\) the employees work. (b) What would be the average number of defective items if the employees average 40 hours per week? (c) According to this relationship, how many hours per week would the employees need to average in order to reduce the average number of defective items to \(500 ?\) Do you think this is a practical goal? Explain.

Round off to the nearest hundredth when necessary. A gourmet food shop sells custom blended coffee by the ounce. Suppose that 1 oz sells for 1.70 dollar and 2 oz sells for 3.20 dollar Assume that the cost \(c\) of the coffee is linearly related to the number of ounces \(n\) purchased (where \(n \geq 1\) ). (a) Write an equation relating the cost of the coffee to the number of ounces purchased. (b) What would be the cost of 3.5 oz of coffee? (c) Suppose a package of coffee is marked at 6.50 dollar but has no indication of how much coffee it contains. Determine the number of ounces of coffee this package contains. (d) Sketch a graph of this equation for \(n \geq 1\) using the horizontal axis for \(n\) and the vertical axis for \(c .\)

Plot the points in the following set: \(\\{(x, y) | y=x+2, \text { and } x=-3,0,4\\}\)
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