Linear Equations, Functions and Their Applications

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the characteristic of linear equations in terms of the power of variables?

Each variable in the equation is raised to the first power. (correct)

The variables can have any power.

The variables can have powers that are multiples of 2.

The variables can have powers that are multiples of 3.

What is the standard form of a linear equation involving two variables x and y?

ax + by = 0

ax - by = C

ax + by = C (correct)

ax / by = C

Which of the following equations is not linear?

x^2 + y = 4 (correct)

2x + 3y = 7

x + 2y = 5

x - 3y = 2

What is the characteristic of linear equations in terms of the product of variables?

The product of variables is never present. Signup and view all the answers

What is the general form of a linear equation involving 'n' variables?

a1x1 + a2x2 + ... + anxn = C Signup and view all the answers

Which of the following equations is linear?

2x - 3y = 4 Signup and view all the answers

What is the first step to determine a pair of values that satisfy a linear equation?

Assume a value for one of the variables Signup and view all the answers

In the equation y = 2x + 500, what is the value of y when x = 20?

520 Signup and view all the answers

What is the purpose of using linear equations in business application problems?

To model real-world problems and make predictions Signup and view all the answers

In the production problem, how many hours of labor are required to produce one unit of product A?

3 hours Signup and view all the answers

What is the total number of hours available for manufacturing products A and B?

120 hours Signup and view all the answers

What is the general form of a linear function in business application problems?

ax + by = C Signup and view all the answers

What is the general form of a linear function involving one independent variable x and a dependent variable y?

y = a1x + a0 Signup and view all the answers

What does the slope 'a1' represent in a linear function?

The rate of change of y with respect to x Signup and view all the answers

What is the weekly salary of the salesperson if they sold 150 units in a week, according to the equation y = 3x + 25?

425 birr Signup and view all the answers

What is the change in the weekly salary associated with each additional unit sold, according to the equation y = 3x + 25?

An increase of 3 birr Signup and view all the answers

What is the fixed cost of production, according to the equation y = 3x + 25?

25 birr Signup and view all the answers

What is the relationship between the change in weekly salary and the change in number of units sold, according to the equation y = 3x + 25?

The change in weekly salary is directly proportional to the change in number of units sold Signup and view all the answers

Study Notes

Linear Equations and their Characteristics

Linear equations are equations whose terms are constant or a constant times one variable to the first power.
Characteristics of linear equations:
- Linear equations are first degree equations, where each variable is raised to the first power.
- The presence of terms with exponents other than 1 excludes an equation from being considered linear.
- Linear equations involve a product of a constant term with a variable, but not a product of variables.
Standard form of a linear equation involving two variables: ax + by = C, where a, b, and C are constants, and a and b cannot both equal zero.

Solving Linear Equations

To determine a pair of values that satisfy a linear equation, assume a value for one of the variables, substitute it into the equation, and solve for the other variable.

Applications of Linear Equations

Example: A company manufactures two products A and B, and has 120 hours of labor available. Each unit of A requires 3 hours of labor, and each unit of B requires 2.5 hours.
Required: Define an equation that states the total work hours used for producing the two units.

Slope of Linear Equations

The slope of a linear equation represents the rate of change of the dependent variable with respect to the independent variable.
Example: Graph the equation 2x + 4y = 16.

Linear Functions and Their Applications

Linear Functions

A linear function has the general form: Y = f(x) = a1x + a0, where a1 and a0 are constants, and a1 ≠ 0.
A change in the value of y is directly proportional to a change in the value of x, with a constant rate of change represented by the slope a1.
Example: Imagine that your weekly salary as a salesperson depends on the number of units you sell each week, following the equation y = 3x + 25.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz covers the basics of linear equations, including their characteristics and applications. Learn about the definition of linear equations and how to identify them.