Problem 63
Question
For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. $$ (2,3) \text { and }(5,7) $$
Step-by-Step Solution
Verified Answer
The slope is \(\frac{4}{3}\) and the line is increasing.
1Step 1: Understand the Slope Formula
The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \(m\) is the slope.
2Step 2: Substitute the Points into the Formula
Use the given points \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (5, 7)\) in the slope formula: \[ m = \frac{7 - 3}{5 - 2} \]
3Step 3: Calculate the Subtraction
Perform the subtraction in the numerator and the denominator: \(7 - 3 = 4\) and \(5 - 2 = 3\).
4Step 4: Divide to Find the Slope
Divide the results from the previous step: \[ m = \frac{4}{3} \] The slope is \(\frac{4}{3}\).
5Step 5: Determine the Line Type
The slope \(\frac{4}{3}\) is positive, indicating the line is increasing.
Key Concepts
Line EquationsPoints and CoordinatesIncreasing and Decreasing Lines
Line Equations
Line equations are fundamental in understanding how lines behave on a graph. A line equation describes the relationship between the x and y coordinates of any point on the line. One of the most common forms is the slope-intercept form, presented as \( y = mx + b \). In this equation, \( m \) represents the slope, and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.
To construct a line equation, knowing the slope and a point on the line is essential. The slope tells us how steep the line is, and the y-intercept tells us where it begins on the y-axis. This form makes it easy to understand a line's direction and starting position at a glance.
Understanding line equations helps in predicting values and understanding trends. For instance, with the given example, once you know the slope is \( \frac{4}{3} \), you can build the equation by substituting into \( y = mx + b \) if needed.
To construct a line equation, knowing the slope and a point on the line is essential. The slope tells us how steep the line is, and the y-intercept tells us where it begins on the y-axis. This form makes it easy to understand a line's direction and starting position at a glance.
Understanding line equations helps in predicting values and understanding trends. For instance, with the given example, once you know the slope is \( \frac{4}{3} \), you can build the equation by substituting into \( y = mx + b \) if needed.
Points and Coordinates
Points and coordinates form the basis of plotting points on a Cartesian plane, which is a two-dimensional graph divided by an x-axis and a y-axis. Each point has a specific location described by a pair of numbers: \((x, y)\). The first number, \(x\), indicates how far the point is along the horizontal axis, and the second number, \(y\), indicates how far it is along the vertical axis.
In the exercise example, the points \((2,3)\) and \((5,7)\) are given. Here, the point \((2,3)\) means moving 2 units to the right and 3 units up from the origin, while \((5,7)\) means moving 5 units to the right and 7 units up. These coordinates help in determining the slope of the line that passes through them.
By identifying and plotting points accurately, one can decipher the relationship between different coordinates, which becomes crucial in understanding the slope and direction of lines.
In the exercise example, the points \((2,3)\) and \((5,7)\) are given. Here, the point \((2,3)\) means moving 2 units to the right and 3 units up from the origin, while \((5,7)\) means moving 5 units to the right and 7 units up. These coordinates help in determining the slope of the line that passes through them.
By identifying and plotting points accurately, one can decipher the relationship between different coordinates, which becomes crucial in understanding the slope and direction of lines.
Increasing and Decreasing Lines
When analyzing lines, identifying if they are increasing or decreasing is key. This is determined by the slope of the line, usually denoted as \( m \).
An increasing line is one where the slope \( m \) is positive, which means that as you move along the line from left to right, the line rises and the y-values increase. For instance, in the given problem, the line has a slope of \( \frac{4}{3} \), which is positive, indicating that the line is increasing.
Decreasing lines, on the other hand, have a negative slope. This means that as you move to the right, the y-values decrease and the line falls. If the calculated slope was negative, the line would be decreasing.
An increasing line is one where the slope \( m \) is positive, which means that as you move along the line from left to right, the line rises and the y-values increase. For instance, in the given problem, the line has a slope of \( \frac{4}{3} \), which is positive, indicating that the line is increasing.
Decreasing lines, on the other hand, have a negative slope. This means that as you move to the right, the y-values decrease and the line falls. If the calculated slope was negative, the line would be decreasing.
- Horizontal lines, where \( m = 0 \), show no change in y-values as you move along the x-axis.
- Vertical lines have an undefined slope since the x-values do not change, making the denominator zero in the slope formula.
Other exercises in this chapter
Problem 62
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