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Chapter 3: Problem 55

Use the slope formula to find the slope of the line that passes through the points. \(\left(0, \frac{3}{4}\right) ;\left(\frac{1}{4}, 0\right)\)

Short Answer

Expert verified
The slope is \( -3 \)

Step by step solution

01

Identify the Coordinates

Recognize the given points: \(x_1, y_1\) and \(x_2, y_2\). In this case: \(x_1 = 0, y_1 = \frac{3}{4}\) and \(x_2 = \frac{1}{4}, y_2 = 0\).
02

Write the Formula for Slope

The slope \(m\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
03

Substitute the Values into the Slope Formula

Plug the coordinates into the formula: \[ m = \frac{0 - \frac{3}{4}}{\frac{1}{4} - 0} \]
04

Simplify the Expression

Simplify the numerator and the denominator: \[ m = \frac{-\frac{3}{4}}{\frac{1}{4}} \]
05

Perform the Division

Divide the fractions: \[ m = -\frac{3}{4} \times \frac{4}{1} = -3 \]

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Key Concepts

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

identify coordinates
Begin by identifying the coordinates given in the exercise. In our case, the points provided are \(0, \frac{3}{4}\) and \(\frac{1}{4}, 0\). You need to label these points correctly with \(x_1, y_1\) and \(x_2, y_2\):
  • \(x_1 = 0\)
  • \(y_1 = \frac{3}{4}\)
  • \(x_2 = \frac{1}{4}\)
  • \(y_2 = 0\)
This step is crucial because it sets up your subsequent calculations. Getting these wrong means all following steps will be wrong.
substitute values
Once you have the coordinates, substitute these values into the slope formula. The slope formula is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Next, replace \(x_1, y_1\), \(x_2, y_2\) with the identified values:
  • \(x_1 = 0\)
  • \(y_1 = \frac{3}{4}\)
  • \(x_2 = \frac{1}{4}\)
  • \(y_2 = 0\)
After substituting, you should get: \[ m = \frac{0 - \frac{3}{4}}{\frac{1}{4} - 0} \] This will help you move into the next step of simplifying the expression.
simplify expression
Now that the values have been substituted into the formula, the next step is to simplify the expression. The expression from the previous step is:
\[ m = \frac{0 - \frac{3}{4}}{\frac{1}{4} - 0} \]
First, simplify both the numerator and the denominator separately:
  • Numerator: \(0 - \frac{3}{4} = -\frac{3}{4}\)
  • Denominator: \(\frac{1}{4} - 0 = \frac{1}{4}\)
This gives you a new expression: \[ m = \frac{-\frac{3}{4}}{\frac{1}{4}} \] By simplifying incrementally, you keep the calculations more manageable and understandable.
perform division
In the final step, you need to perform the division of the simplified expression. The expression you have is:
\[ m = \frac{-\frac{3}{4}}{\frac{1}{4}} \]
To divide these fractions, multiply by the reciprocal of the denominator:
\[ m = -\frac{3}{4} \times \frac{4}{1} \]
Simplifying the multiplication: \[ m = -3 \]
So, the slope of the line passing through the points \(0, \frac{3}{4}\) and \(\frac{1}{4}, 0\) is \(m = -3\). This method shows how understanding each component of the slope formula leads to accurate results.

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Most popular questions from this chapter

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(y=-x+2\)

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The amount of fresh tomatoes consumed per person in the United States increased from \(89.9\) lb in 2009 to 93.5 lb in 2011. (Source: www.ers.usda.gov, Dec. 15,2011\()\)

Use the slope formula to find the slope of the line that passes through the points. \((3,11) ;(7,39)\)

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(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-20,-50) ;(10,30)\)
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