Problem 24
Question
In Problems 23-28, find the slope of the line containing the given two points. \((3,5)\) and \((4,7)\)
Step-by-Step Solution
Verified Answer
The slope of the line is 2.
1Step 1: Understand the Slope Formula
The slope of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula represents the change in \( y \) (vertical change) divided by the change in \( x \) (horizontal change) between the two points.
2Step 2: Identify the Coordinates
Take the coordinates from the given points: \( (3, 5) \) which implies \( x_1 = 3 \) and \( y_1 = 5 \) and \( (4, 7) \) which implies \( x_2 = 4 \) and \( y_2 = 7 \). These are the values we will substitute into the slope formula.
3Step 3: Substitute the Values into the Formula
Insert the identified coordinates into the slope formula: \[ m = \frac{7 - 5}{4 - 3} \] Here, \( y_2 \) is 7, \( y_1 \) is 5, \( x_2 \) is 4, and \( x_1 \) is 3.
4Step 4: Perform the Calculations
Calculate the differences in the numerator and the denominator: \[ \text{Change in } y = 7 - 5 = 2 \] \[ \text{Change in } x = 4 - 3 = 1 \] Thus, the slope \( m = \frac{2}{1} \).
5Step 5: Simplify the Slope
Simplify \( \frac{2}{1} \) to get the slope: \[ m = 2 \] This is the slope of the line passing through the points (3, 5) and (4, 7).
Key Concepts
Slope FormulaCoordinate GeometryLinear Equations
Slope Formula
The slope formula is a fundamental concept in coordinate geometry used to determine the steepness or incline of a line on a graph. It is represented mathematically as \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line. The numerator \( y_2 - y_1 \) reflects the vertical change (also known as the "rise"), while the denominator \( x_2 - x_1 \) captures the horizontal change (or "run") between these points.
The slope tells us how much "up" or "down" movement there is for every unit of "across" movement. A positive slope suggests an upward direction, a negative slope indicates a downward trend, zero slope depicts a horizontal line, and an undefined slope arises in a vertical line situation. Understanding this formula is crucial for exploring more complex linear relationships.
The slope tells us how much "up" or "down" movement there is for every unit of "across" movement. A positive slope suggests an upward direction, a negative slope indicates a downward trend, zero slope depicts a horizontal line, and an undefined slope arises in a vertical line situation. Understanding this formula is crucial for exploring more complex linear relationships.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, marries algebra and geometry using a coordinate system. It allows us to assign coordinates, typically in an \( xy \)-plane, to express geometric figures numerically and algebraically. This system provides a way to examine line positions, properties, and interrelationships through equations.
Points in coordinate geometry are depicted as ordered pairs \( (x, y) \), where \( x \) represents the horizontal axis (abscissa) and \( y \) stands for the vertical axis (ordinate). In our exercise, points \( (3, 5) \) and \( (4, 7) \) were utilized to calculate the line's slope. Using the coordinate plane simplifies understanding geometric properties, making it easier to solve real-world problems that involve space and position.
Points in coordinate geometry are depicted as ordered pairs \( (x, y) \), where \( x \) represents the horizontal axis (abscissa) and \( y \) stands for the vertical axis (ordinate). In our exercise, points \( (3, 5) \) and \( (4, 7) \) were utilized to calculate the line's slope. Using the coordinate plane simplifies understanding geometric properties, making it easier to solve real-world problems that involve space and position.
- Provides a framework for describing geometric shapes with algebraic expressions
- Useful for visualizing relationships and solving equations graphically
- Facilitates the calculation of distances, midpoints, and slopes
Linear Equations
Linear equations are equations of the first order, represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The line generated by plotting this equation on a graph is straight. It showcases a constant rate of change, which the slope embodies. Linear equations are a significant component of algebra and have wide applications—in economic models, physics, engineering, among others.
From the given points \( (3,5) \) and \( (4,7) \), after calculating the slope as \( m = 2 \), a linear equation can be formed. First, substitute one of the points and the slope into the slope-intercept form \( y = mx + b \) to find \( b \). For example, using \( (3,5) \):
From the given points \( (3,5) \) and \( (4,7) \), after calculating the slope as \( m = 2 \), a linear equation can be formed. First, substitute one of the points and the slope into the slope-intercept form \( y = mx + b \) to find \( b \). For example, using \( (3,5) \):
- \( 5 = 2 \times 3 + b \)
- Solve for \( b \) to get \( b = -1 \)
Other exercises in this chapter
Problem 23
Perform the indicated operations and simplify. \(\frac{x^{2}-4}{x-2}\)
View solution Problem 24
Which of the following represent the same graph? Check your result analytically using trigonometric identities. (a) \(y=\sin \left(x+\frac{\pi}{2}\right)\) (b)
View solution Problem 24
$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{5}(x+3)-\log _{5} x=1 $$
View solution Problem 24
Is every polynomial of even degree an even function? Is every polynomial of odd degree an odd function? Explain.
View solution