Understanding Equations and Slope of a Straight Line

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Questions and Answers

What does the slope of a line represent?

The steepness of the line

The y-intercept of the line

The x-intercept of the line

The rate at which the line raises or falls (correct)

If a line has a slope of 0, what type of line is it?

Vertical line

Positive slope line

Horizontal line (correct)

Negative slope line

Which formula is used to calculate the slope between two points on a straight line?

$\frac{x_2 + x_1}{y_2 + y_1}$

$\frac{y_2 - y_1}{x_2 - x_1}$ (correct)

$\frac{x_2 - x_1}{y_2 - y_1}$

$\frac{y_2 + y_1}{x_2 + x_1}$

If a line has a positive slope, how does it behave from left to right?

It rises Signup and view all the answers

In the slope calculation formula (mathematically), what do (x1, y1) and (x2, y2) represent?

(x1, y1) and (x2, y2) coordinates respectively Signup and view all the answers

What does it mean if a line has an undefined slope?

It is a vertical line Signup and view all the answers

What is the slope-intercept form of an equation where the slope is denoted by 𝑚 and the y-intercept is denoted by 𝑘?

𝑦 = 𝑚𝑥 + 𝑘 Signup and view all the answers

In the equation 𝑦 = −5𝑥 + 15, what is the slope of the line?

-5 Signup and view all the answers

If a line has a slope of -2 and passes through the point (2, 8), what is the equation of this line in slope-intercept form?

𝑦 = -2𝑥 + 12 Signup and view all the answers

If a line has a slope of 0 and a y-intercept of 4, what is the equation of this line?

𝑦 = 0 + 4𝑥 Signup and view all the answers

For a non-vertical straight line with slope 𝑚 and containing point (3, 5), what is the slope of the line connecting (3, 5) with any other point (7, 9)?

-2 Signup and view all the answers

In the equation 𝑎𝑥 + 𝑏𝑦 = 𝑐, how can you rewrite it to be in slope-intercept form?

$y = \frac{-a}{b}x + \frac{c}{b}$ Signup and view all the answers

What does a slope of 3 indicate in a linear function?

The increase in weekly salary associated with each additional unit sold Signup and view all the answers

In a linear function with two variables x and x_2, what does the equation y = f(x, x_2) = ax + ax_2 + a represent?

The dependent variable y as a function of x and x_2 Signup and view all the answers

What is the two-point formula for calculating the slope of a line?

Slope = (y2 - y1) / (x2 - x1) Signup and view all the answers

In a linear cost function, why are organizations concerned with costs?

Costs reflect the money flowing out of the organization Signup and view all the answers

How does a linear function y = 300x + 500x_2 + 2500 calculate the weekly salary for a salesperson?

By multiplying the number of units sold by respective commission values Signup and view all the answers

What does it mean when a linear function states that 'a change in the value of y is directly proportional to a change in the variable x'?

Increasing x will always increase y Signup and view all the answers

Study Notes

Equations of Lines

Equations of the form x = k have no y-intercept, and equations of the form y = k have no x-intercept.

Slope of a Line

Slope represents the inclination of a line or the rate at which the line rises and falls.
The slope of a line can be positive, negative, zero, or undefined.
A positive slope indicates a line that rises from left to right.
A negative slope indicates a line that falls from left to right.
A zero slope indicates a horizontal line.
An undefined slope indicates a vertical line.
The sign of the slope shows whether the line is falling or rising, and its magnitude shows the steepness of the line.

Two-Point Formula

The slope of a line can be computed using the two-point formula: Slope = Δy / Δx.
The formula for the slope of a line connecting two points (x1, y1) and (x2, y2) is: Slope = (y2 - y1) / (x2 - x1).

Slope-Intercept Form

The slope-intercept form of a linear equation is: y = mx + k, where m is the slope and k is the y-intercept.
The slope-intercept form can be derived from the general form of a linear equation: ax + by = c.

Determining the Equation of a Straight Line

If the slope and y-intercept are given, the equation of the line can be written as: y = mx + k.
If the slope and one point are given, the equation of the line can be written by substituting the known slope and coordinates of the point into the slope-intercept form.

Linear Functions

A linear function has the form: y = f(x) = ax + a.
A linear function indicates that a change in the value of y is directly proportional to a change in the value of x.
Linear functions can have multiple independent variables, such as y = f(x, x2) = ax + ax2 + a.

Linear Cost Function

Linear cost functions represent the costs associated with an organization, which reflect the money flowing out of the organization.
Linear cost functions can be used to model the costs of producing and selling products.

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Description

Learn about how equations of the form x=k and y=k impact the y-intercept and x-intercept, as well as the concept of slope and its significance in characterizing straight lines. Understand how slope reflects the inclination and steepness of a line, with examples of positive, negative, zero, and undefined slopes.