Problem 63 - Algebra 1: Concepts and Skills

Question

The selling price of a certain video is \(\$ 7\) more than the price the store paid. If the selling price is \(\$ 24,\) find the equation that determines the price the store paid. $$ \begin{array}{llll} {\text { (A) } x+7=24} & {\text { (B) } x=7+24} & {\text { (C) } 7-24=x} & {\text { (D) } x=24} \end{array} $$

Step-by-Step Solution

Answer

The correct equation that determines the price the store paid for the video is (A) \(x + 7 = 24\).

Step 1: Identify the Relevant Information

The exercise tells us that the selling price (S) is $24. This is $7 more than the price the store paid (P). It's represented mathematically as: \(S = P + 7\). Given that \(S = 24\), we can substitute this into the equation. This will give us our equation to find the price the store paid.

Step 2: Substitute Seller Price in the Equation

By substituting the value of S, we get: \(24 = P + 7\). This represents the price the store paid for the video.

Step 3: Match with Given Options

When we compare our derived equation with the given options, it precisely matches with option (A) i.e., \(x + 7 = 24\). Hence, option (A) is the correct equation that determines the price the store paid for the video.

Key Concepts

Selling Price

When we talk about selling price, we're referring to the amount of money a customer pays to purchase an item. In the given exercise, the selling price of the video is $24. The selling price is an important concept in commerce because it directly affects profit. For a video or any item to be sold successfully, the price must be appealing enough for a customer to consider buying it. However, it must also be set in a way that ensures the store makes a profit.

• A competitive selling price considers costs, demand, and competition.
• A well-set selling price should cover the store's expenses.
• It should also include a profit margin.

Understanding the selling price aids in creating effective sales strategies that maximize sales and profit.

Store Price

The store price, also known as the cost price, is the amount the store originally paid to acquire the item for resale. In this problem, we need to find out the store price for the video given two pieces of information: the selling price and the additional amount added on to it, which is $7 in this case.

• The equation reflects that the selling price is higher than the store price due to the added margin.
• The store price is critical because if it's set correctly, and alongside the selling price, it ensures the store makes a profit.
• Once expenses are covered, any remaining amount contributes to profit.

The goal is to determine the initial price at which the store acquired the video, which involves using an equation to deduce this from the given selling price.

Substitution Method

The substitution method is an algebraic technique used to solve equations, where we substitute known values into an equation to find unknowns. In this exercise, it involves inserting the given selling price into an equation to find the store price. Here's how it works:

• You start with an algebraic equation that relates two or more values.
• Identify the known and unknown variables.
• Substitute the known value into the equation. In this case, the selling price $24 is substituted into the equation as the value of S, yielding 24 = P + 7.

This process simplifies the problem since you replace complex expressions with known numbers, making it easier to isolate the unknown variable. Here, the solution shows that by substituting, we rearrange the equation to solve for P, which allows us to find the store price easily. Substitution is a critical skill in algebra used to simplify solving mathematical problems. By practicing this method, students become more proficient in solving different forms of equations.