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 price the store paid.

Short answer

The store paid \$17 for the video.

Step-by-step solution

Understand the problem and translate into algebra

First, we need to express the given problem in an algebraic equation. We can let the variable \(x\) represent the price the store paid. Consequently, since the selling price is \$7 more than the price the store paid, we can represent the selling price as \(x + 7 \).Given that the selling price is \$24, we equate \(x + 7 = 24 \).

Solving the equation

We need to solve the equation for \(x\), this will give us the price the store paid. To get \(x\) alone on the left side of the equation, we subtract 7 from both sides of the equation. Doing this, \(x = 24 - 7 \).

Getting the final answer

Now, we just perform the subtraction operation on the right side to get the value of \(x\). Therefore, \(x = 17 \).

Key concepts

Solving Algebraic Equations

Solving algebraic equations is a fundamental skill in mathematics that allows us to find unknown values, often represented by variables like x or y. These equations are like puzzles, and by applying certain rules, we can figure out the value of these unknowns.

For example, consider the equation from the exercise: x + 7 = 24. This is a simple linear equation where x represents the price the store paid. To solve for x, we need to isolate the variable on one side of the equation. We do this by performing the same operation on both sides of the equation to maintain the balance, known as the equilibrium of the equation.

Subtraction Rule in Equation Solving

As shown in the step-by-step solution, subtracting 7 from both sides of the equation x + 7 = 24 gives us x = 24 - 7. This step is crucial because it simplifies the equation, making the variable x the subject and thus computable. Finally, when we calculate 24 - 7, we discover that x = 17, which is the price the store paid for the video. To master algebraic equations, practice is key, and being meticulous with each step ensures fewer mistakes and a greater understanding of the concept.

Linear Equations

Linear equations are the simplest type of equations we encounter in algebra. They involve variables that are raised to no higher power than one and can be easily visualized as a straight line on a graph. Generally, they are written in the form ax + b = c, where a, b, and c are constants, and x is the variable we are solving for.

Our exercise involved a basic linear equation. It is linear because the variable x is to the first power, which means its graph is a straight line.

Importance of Linear Equations

Understanding linear equations is important because they model many real-world scenarios. These scenarios can range from calculating budgets to predicting profits. They also form a foundation for more complex types of algebraic equations. The principles used in solving them, like balancing and simplifying, apply to more complicated algebra problems as well.

Algebra Word Problems

Algebra word problems require us to translate scenarios from words to mathematical expressions. These problems help develop critical thinking and problem-solving skills because you can't solve what you don't understand.

In the given exercise, the word problem described a real-life situation involving the selling price and cost price of a video. The key to solving such problems is to identify the unknown quantities and define them with variables, then translate the rest of the words into a solvable algebraic equation.

Steps to Solve Word Problems

The first step is comprehension: read the problem carefully and determine what is being asked. Next, identify and assign a variable to the unknown quantity—it was x in our example. After that, construct an equation based on the relationships described in the problem. Following these steps ensures a structured approach to solving word problems that can be applied to a variety of scenarios in both academic settings and everyday life.