Problem 105
Question
Use the results of Exercises 101–104 to write an equation of the line that passes through the points. \(x\) -intercept: (2,0) \(y\) -intercept: (0,9)
Step-by-Step Solution
Verified Answer
The equation of the line that passes through the points (2,0) and (0,9) is \(y = -4.5x + 9\).
1Step 1: Calculate the Slope
To calculate the slope of the line, use the formula for slope which is \(m = (y_2 - y_1) / (x_2 - x_1)\). Substituting the given points into the formula gives, \(m = (9 - 0) / (0 - 2) = -9 / 2 = -4.5\).
2Step 2: Write the equation
The general form of equation of a line is \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept. Substitute the calculated slope and known y-intercept into the equation. This results in \(y = -4.5 x + 9\).
Key Concepts
Slope CalculationY-interceptLinear Equations in Algebra
Slope Calculation
The slope of a line is a measure of its steepness and is a central concept in the study of linear equations in algebra. It's typically represented as 'm' in the equation of a line, which can be in the form of
For the exercise given, the slope is calculated using the
y = mx + b. The slope can be calculated by taking any two points on the line, and dividing the difference in their y-coordinates by the difference in their x-coordinates. The formula for this is m = (y_2 - y_1) / (x_2 - x_1).For the exercise given, the slope is calculated using the
x-intercept (2,0) and the y-intercept (0,9). Plugging these points into the slope formula, we get m = (9 - 0) / (0 - 2) = 9 / -2 , which simplifies to m = -4.5. Remember, a positive slope means the line goes upward as it moves from left to right, while a negative slope means it goes downward. This latter case applies here, indicating a line that descends as it progresses to the right.Y-intercept
The y-intercept of a line is the point where the line crosses the y-axis of the coordinate system. This happens when the
Based on the exercise, the y-intercept has been provided as the point (0,9). This means that when
x-value is zero. In the equation form y = mx + b, the b represents the y-intercept. It's an important feature of the line because it provides a quick reference point for the line's position in the coordinate plane.Based on the exercise, the y-intercept has been provided as the point (0,9). This means that when
x = 0, y = 9. Therefore, in the context of the provided exercise, b = 9. This information is vital as it helps anchor the line at the right position vertically in the coordinate plane. It's one of two essential characteristics needed to completely describe a line's equation in the slope-intercept form.Linear Equations in Algebra
Linear equations form the foundation of algebra and graphing. They can describe a straight line in a coordinate plane using an equation that includes both the slope and y-intercept. The standard format is the slope-intercept form, expressed as
In this form, the equation effortlessly communicates the line's behavior: its steepness, its direction, and where it intersects the y-axis. Given the slope and y-intercept from the exercise,
y = mx + b, where m is the slope, and b is the y-intercept.In this form, the equation effortlessly communicates the line's behavior: its steepness, its direction, and where it intersects the y-axis. Given the slope and y-intercept from the exercise,
m = -4.5 and b = 9, the equation for the line becomes y = -4.5x + 9. The negative slope indicates a line decreasing from left to right, and the y-intercept places the line at the correct vertical starting point. Students can plot this equation on a graph by starting at the y-intercept and using the slope to find another point, thus drawing a complete line.Other exercises in this chapter
Problem 104
Use a graphing utility to graph the equation of the line in the form $$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0$$ Use the graphs to make a conjecture
View solution Problem 105
Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$\left(f^{-1} \circ g^{-1}\right)(1)$$
View solution Problem 106
Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$\left(g^{-1} \circ f^{-1}\right)(-3)$$
View solution Problem 106
Use the results of Exercises 101–104 to write an equation of the line that passes through the points. \(x\) -intercept: (-5,0) \(y\) -intercept: (0,-4)
View solution