Problem 52 - 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 \(\$ 50\). Write an equation that models only your income.

Step-by-Step Solution

Answer

The equation that models only the income from drawing sketches is \(I = 12x\).

Step 1: Identify the Variables

In this case, the number of sketches drawn can be represented by the variable \(x\), and the income per sketch is \(\$12\). The overall income can be represented by the variable \(I\).

Step 2: Formulate the Equation

Since the income is given by the product of the number of sketches and the income per sketch, the equation modelling the income can be written as, \(I = 12x\).

Step 3: Write the Final Equation

The final equation representing the income only, according to the given conditions can therefore be written as \(I = 12x\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is essential for solving problems in algebra. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables are symbols used to represent unknown values. In the context of our example with sketches, the money earned from each sketch, which is a known constant value of \(\$12\), and the expenses for art supplies, a fixed \(\$50\), are considered constants in the algebraic expression. To form an algebraic expression for this scenario, you multiply the number of sketches (represented by a variable) by the income per sketch (a constant).

For the income situation in our exercise, the algebraic expression is written as, \(I = 12x\), wherein \(I\) represents the total income from drawing portraits, and \(x\) is the number of sketches drawn. This expression does not include the initial expense for art supplies because the problem only asks for an equation that models the income.

Variable Representation

In algebra, variables are symbols that stand in for unknown or changeable values. They are the backbone of equation formulation and are commonly denoted by letters such as \(x\), \(y\), or \(z\). In the given problem, the variable \(x\) is used to represent the number of sketches drawn, which is a value we don't know beforehand. The importance of variable representation lies in its ability to generalize the problem so that the equation can be applied to any number of sketches.

When constructing an equation or expression, the correct representation of variables is crucial. If we change the variable \(x\) to \(y\), for instance, our algebraic expression becomes \(I = 12y\). Although this change does not affect the mathematical validity, consistency in variable representation helps to avoid confusion in more complex problems.

Equation Formulation

Equation formulation is the process of translating a word problem or situation into a mathematical equation. It involves assigning variables, determining relationships between them, and using appropriate mathematical operations. It's like writing a recipe for the mathematical problem - one must include all the right ingredients (variables and constants) and follow the correct sequence of steps (operations).

In our example, there are two crucial steps to formulating the correct equation for the income earned from drawing sketches. First, identify the relevant variables: the number of sketches (\(x\)) and total income (\(I\)). Then, ascertain the relationship between these variables: income is obtained by multiplying the number of sketches by the income earned per sketch. This gives us the simple equation \(I = 12x\), which directly models the income based on the number of sketches drawn, without incorporating unrelated factors such as the fixed expense for art supplies.