Problem 4
Question
Find the slope of the line through \(P\) and \(Q .\) \(P(1,2), Q(3,3)\)
Step-by-Step Solution
Verified Answer
The slope of the line is \( \frac{1}{2} \).
1Step 1: Identify Coordinates
We are given two points, \( P(1, 2) \) and \( Q(3, 3) \). Each point has an \( x \)-coordinate and a \( y \)-coordinate. For point \( P \), \( x_1 = 1 \) and \( y_1 = 2 \). For point \( Q \), \( x_2 = 3 \) and \( y_2 = 3 \).
2Step 2: Understand the Slope Formula
The formula to find the slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula calculates the vertical change \( y_2 - y_1 \) divided by the horizontal change \( x_2 - x_1 \).
3Step 3: Calculate the Vertical Change
Calculate \( y_2 - y_1 \): \( 3 - 2 = 1 \). This is the vertical change between points \( P \) and \( Q \).
4Step 4: Calculate the Horizontal Change
Calculate \( x_2 - x_1 \): \( 3 - 1 = 2 \). This is the horizontal change between points \( P \) and \( Q \).
5Step 5: Compute the Slope
Using the slope formula, substitute the vertical and horizontal changes: \( m = \frac{1}{2} \). This means the slope of the line through points \( P \) and \( Q \) is \( \frac{1}{2} \).
Key Concepts
Coordinate GeometryLinear EquationsSlope Formula
Coordinate Geometry
Coordinate geometry is a branch of mathematics dealing with the study of geometry using a coordinate system. This branch is crucial because it allows us to describe geometric shapes numerically and extract necessary information using algebra. In more simple terms, coordinate geometry involves the use of graphs and coordinates to solve geometric problems.
When working with coordinate geometry, we often start by identifying coordinates on a plane. Points such as \( P(1, 2) \) and \( Q(3, 3) \) are defined by their positions on the grid using an \( x \)-coordinate and a \( y \)-coordinate. These ordered pairs help us understand positions relative to each other and calculate important aspects like distances, midpoints, and slopes.
When working with coordinate geometry, we often start by identifying coordinates on a plane. Points such as \( P(1, 2) \) and \( Q(3, 3) \) are defined by their positions on the grid using an \( x \)-coordinate and a \( y \)-coordinate. These ordered pairs help us understand positions relative to each other and calculate important aspects like distances, midpoints, and slopes.
- The \( x \)-coordinate (abscissa) represents the horizontal distance from the origin.
- The \( y \)-coordinate (ordinate) shows the vertical distance from the origin.
Linear Equations
Linear equations are equations that produce a straight line when graphed. They have a simple form, generally deemed easy to work with. These equations often appear as \( y = mx + c \), where \( m \) is the slope of the line and \( c \) is the y-intercept, which is the point where the line crosses the y-axis.
The ability to understand and solve linear equations is essential because they are fundamental in algebra and calculus. The concept of a linear equation helps us model real-world situations where relationships are linear. For example, the equation \( y = \frac{1}{2}x + b \) in our problem would form a straight line with a slope of \( \frac{1}{2} \), as calculated from points \( (1, 2) \) and \( (3, 3) \).
The ability to understand and solve linear equations is essential because they are fundamental in algebra and calculus. The concept of a linear equation helps us model real-world situations where relationships are linear. For example, the equation \( y = \frac{1}{2}x + b \) in our problem would form a straight line with a slope of \( \frac{1}{2} \), as calculated from points \( (1, 2) \) and \( (3, 3) \).
- Linear equations can represent relationships with constant rates or proportionality.
- They are useful in predictive modeling, analyzing trends, and finding intersections of lines (system of equations).
Slope Formula
The slope formula is a straightforward yet extremely useful tool in coordinate geometry that helps us calculate the steepness or incline of a line. The slope, often denoted by \( m \), is calculated using the following formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula tells us how much the \( y \)-coordinate changes for a given change in the \( x \)-coordinate between two points on a line.
To find the slope from a set of points, you subtract the \( y \)-coordinate of the first point from the \( y \)-coordinate of the second, and the \( x \)-coordinate of the first from the \( x \)-coordinate of the second. This calculation gives you a ratio of the change in \( y \) to the change in \( x \), showing how quickly the line rises or falls as we move along it.
This formula tells us how much the \( y \)-coordinate changes for a given change in the \( x \)-coordinate between two points on a line.
To find the slope from a set of points, you subtract the \( y \)-coordinate of the first point from the \( y \)-coordinate of the second, and the \( x \)-coordinate of the first from the \( x \)-coordinate of the second. This calculation gives you a ratio of the change in \( y \) to the change in \( x \), showing how quickly the line rises or falls as we move along it.
- A positive slope means the line moves upwards as we move from left to right.
- A negative slope means the line moves downwards as we move from left to right.
- A slope of zero indicates a horizontal line, showing no vertical change.
Other exercises in this chapter
Problem 3
1–6 ? Determine whether the given points are on the graph of the equation. $$ x-2 y-1=0 ; \quad(0,0),(1,0),(-1,-1) $$
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1–12 ? Write an equation that expresses the statement. \(w\) is jointly proportional to \(m\) and \(n\)
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Use a graphing calculator or computer to decide which viewing rectangle \((\mathrm{a})-(\mathrm{d})\) produces the most appropriate graph of the equation. $$ \b
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1–6 ? Determine whether the given points are on the graph of the equation. $$ y\left(x^{2}+1\right)=1 ; \quad(1,1),\left(1, \frac{1}{2}\right),\left(-1, \frac{1
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