Business Mathematics Review

Flashcards covering key concepts from the Business Mathematics Review, including linear equations, break-even analysis, systems of equations, and matrix operations.

How do you find the equation of a line that passes through two given points (x₁, y₁) and (x₂, y₂)?

First, calculate the slope m = (y₂ - y₁) / (x₂ - x₁). Then, use the point-slope form y - y₁ = m(x - x₁) and simplify to the slope-intercept form y = mx + b.

How do you find an equation of a line given its slope (m) and y-intercept (b)?

Use the slope-intercept form of a linear equation, y = mx + b.

What is the slope of a vertical line?

The slope of a vertical line is undefined.

What is the equation of a horizontal line that passes through the point (a, b)?

The equation of a horizontal line is y = b.

How is the break-even point determined for a firm when its cost function C(x) and revenue function R(x) are given?

The break-even point occurs when total cost equals total revenue (C(x) = R(x)). Solve this equation for x (number of units) and then calculate R(x) or C(x) for the break-even revenue.

How do you calculate the break-even production and revenue given selling price (p), variable cost as a percentage of selling price, and monthly fixed costs (F)?

The variable cost per unit (v) is (percentage variable cost) * p. The cost function is C(x) = vx + F, and the revenue function is R(x) = px. Set C(x) = R(x) to find the break-even production (x), then calculate R(x) for break-even revenue.

How do you find the equilibrium quantity and equilibrium price in a market?

Build linear equations for both the demand and supply functions. The equilibrium occurs at the point where the quantity demanded equals the quantity supplied, which is the intersection point of the demand and supply curves.

How do you find the linear equation expressing an asset's book value (V) at the end of t years, given its initial cost (C₀), scrap value (S) at the end of 'n' years?

Calculate the annual depreciation as (C₀ - S) / n. The book value equation is V(t) = C₀ - [(C₀ - S) / n] * t.

What are the possible types of solutions for a linear system of equations?

A linear system can have a unique solution, infinitely many solutions, or no solution (inconsistent system).

How do you write the augmented matrix corresponding to a given system of linear equations?

Arrange the coefficients of the variables and the constants from each equation into a matrix, separating the coefficients from the constants with a vertical line.

How can a word problem involving two unknown quantities and two conditions be solved using a system of linear equations?

Define variables for the unknown quantities, set up two linear equations based on the given conditions, and then solve the system using substitution, elimination, or matrix methods.

How do you interpret an augmented matrix in row-reduced form to determine the solution(s) to a system of linear equations?

If there's a row like [0 0 … 0 | k] where k ≠ 0, there is no solution. If the number of free variables is zero, there's a unique solution. If there are free variables, there are infinitely many solutions, express the dependent variables in terms of the free variables.

What is the Gauss-Jordan elimination method used for in linear algebra?

It's a method used to solve systems of linear equations by transforming the augmented matrix of the system into row-reduced echelon form, from which the solution(s) can be directly read.

How is the size (or dimension) of a matrix determined?

The size of a matrix is given by m x n, where m is the number of rows and n is the number of columns.

How do you identify a specific element in a matrix?

An element is identified by its position (row i, column j), denoted as aᵢⱼ.

How are matrix addition and subtraction performed?

To add or subtract matrices, they must have the same dimensions. The operation is performed element-wise, adding or subtracting corresponding elements.

How do you solve for unknown variables in a matrix equation?

Equate the corresponding elements of the matrices on both sides of the equation and solve the resulting system of algebraic equations.

What is the transpose of a matrix?

The transpose of a matrix A, denoted Aᵀ, is obtained by interchanging the rows and columns of A. The element at (i, j) in A becomes the element at (j, i) in Aᵀ.

How is the product of two matrices, A and B, computed?

For the product AB to be defined, the number of columns in A must equal the number of rows in B. The element in the i-th row and j-th column of AB is the dot product of the i-th row of A and the j-th column of B.

How can a system of linear equations be used to solve real-world allocation problems, such as diet planning?

Represent quantities of ingredients as variables and set up equations based on nutritional requirements (proteins, carbohydrates, iron) per unit of each ingredient. Solve the system to find the optimal amounts of each ingredient.

How do you apply a given mathematical formula, such as Cowling's rule for pediatric dosage?

Identify all the variables in the formula, substitute the given values for those variables, and then perform the necessary calculations to find the unknown quantity.

How can matrix multiplication be used to calculate total profit in a business scenario involving multiple product models and locations?

Represent the number of units of each model in each state as one matrix and the profit per unit for each model as a profit vector. Multiplying these matrices (if dimensions align correctly) can yield a matrix showing the total profit for each state.