Problem 53 - Algebra 1

Question

DRAWING PORTRATTS In Exercises \(52-54\), you are earning extra money drawing sketches for your friends and family. You receive \(\$ 12\) per sketch. Your expenses for art supplies are \(\$ 50\). Suppose you plan to spend an additional \(\$ 7\) per sketch for a frame. Write an equation that models only your expenses.

Step-by-Step Solution

Answer

The equation that models the expenses is \( E = 7x + 50 \)

Step 1: Identifying Fixed and Variable Costs

Firstly, sift out the variations of cost in the given problem. We have a fixed cost, which does not change with the number of sketches produced, which is the art supplies cost of $50. Then, we have a variable cost that changes with each sketch made, which is the frame cost of $7 per sketch.

Step 2: Creating the Equation

The next task is to put these costs into a mathematical equation. Let's represent the number of sketches by 'x'. Multiply the cost per sketch ($7) by 'x', then add the fixed cost ($50). Therefore, the equation representing the expenses is E = 7x + 50, where E represents the total expenses.

Step 3: Interpretation

Thereby, the equation provided represents the total expenses incurred to draw and frame 'x' number of sketches. The cost increases by $7 for each sketch due to the frame's cost, and the additional static $50 represents the cost of the art supplies.

Key Concepts

Understanding Fixed and Variable Costs

When managing a small business or a side hustle, such as drawing portraits, it's essential to distinguish between fixed and variable costs. Fixed costs, like a one-time purchase of art supplies, remain constant regardless of how much you produce or sell. It's a set cost that doesn't change with the level of output, symbolized in our example by the \(50 spent on art supplies.

On the other hand, variable costs fluctuate with the production level. The more items you produce or services you perform, the higher these costs will climb. In the portrait drawing scenario, buying frames represents a variable cost; for every new sketch, an additional \)7 is spent on framing. Variable costs can be easily remembered as they 'vary' with the amount of work you do.

Crafting Algebraic Expressions

The language of algebra makes it possible to model real-life situations with algebraic expressions. Creating these expressions involves translating the scenario into mathematical terms. Let's simplify the process for the portrait sketch example. Assign a variable, such as 'x', to represent the number of sketches. This allows you to express the variable cost as '7x', where the 7 represents the frame cost for each sketch, then easily adjust for different quantities of sketches.

An algebraic expression combines numbers, variables, and operation symbols to represent a quantity. In the context of a linear equation, the expression often takes the form of 'mx + b', where 'm' represents a variable cost per unit, 'x' is the number of units, and 'b' represents a fixed cost. This format is particularly useful for understanding how the costs of a business can be encapsulated in a clear and calculable way.

Linear Models in Action

Linear models are powerful tools for representing situations where there is a constant rate of change, such as the cost scenario in the portrait drawing business. The equation 'E = 7x + 50' is a perfect illustration of a linear model. In this equation, 'E' stands for total expenses, while 'x' symbolizes the quantity of sketches drawn. The number 7 reflects the variable cost per sketch, and 50 signifies the fixed cost.

To visualize a linear model, think of plotting a graph with two axes. One axis represents the number of sketches and the other represents the total expense. As the number of sketches increases, the total expense will rise in a straight line, hence the term 'linear.' This linear model is not only easy to interpret, but it also allows you to predict expenses for any number of sketches, making it an invaluable tool for financial planning within any business model.