Problem 67 - Algebra 2

Question

MONEY Allison is saving money to buy a video game system. In the first week, her savings were \(\$ 8\) less than \(\frac{2}{5}\) the price of the system. In the second week, she saved 50 cents more than \(\frac{1}{2}\) the price of the system. She was still \(\$ 37\) short. Find the price of the system.

Step-by-Step Solution

Answer

The price of the video game system is $295.

Step 1: Translate the Problem into an Equation

Let the price of the video game system be \( x \). In the first week, Allison saves \( \frac{2}{5}x - 8 \) dollars. In the second week, she saves \( \frac{1}{2}x + 0.5 \) dollars. We know that the total amount she has is \( x - 37 \), because she is still \$37 short. We can write the equation as: \( (\frac{2}{5}x - 8) + (\frac{1}{2}x + 0.5) = x - 37 \).

Step 2: Combine Like Terms

Combine the terms on the left side of the equation to simplify: \[\frac{2}{5}x - 8 + \frac{1}{2}x + 0.5 = \frac{2}{5}x + \frac{1}{2}x - 7.5\].

Step 3: Clear Fractions

To eliminate the fractions, find a common denominator for \( \frac{2}{5} \) and \( \frac{1}{2} \), which is 10. Multiply every term by 10: \[10(\frac{2}{5}x) + 10(\frac{1}{2}x) - 10 \times 7.5 = 10(x - 37)\]. Thus, \[4x + 5x - 75 = 10x - 370\].

Step 4: Simplify the Equation

Combine the like terms: \[9x - 75 = 10x - 370\].

Step 5: Isolate the Variable

Rearrange the terms to isolate \( x \): \[9x - 10x = -370 + 75\]. This simplifies to \[-x = -295\].

Step 6: Solve for the Price

Divide both sides by -1 to find the value of \( x \): \[x = 295\]. So, the price of the video game system is \$295.

Key Concepts

Solving Equations with Fractions

When solving algebraic equations involving fractions, it’s common to start by eliminating the fractions for simplicity. Fractions can make equations seem more complex, but with a clear process, you can handle them easily. Here’s how you tackle these problems:

• Identify the Fractions: Begin by spotting all the fractional terms in your equation. In our problem, the terms \( \frac{2}{5}x \) and \( \frac{1}{2}x \) are fractions.
• Find a Common Denominator: In order to clear the fractions, find the least common multiple of the denominators. Here, the common denominator between 5 and 2 is 10.
• Multiply Through: To eliminate the fractions, multiply every term of the equation by this common denominator. This transforms our equation into one with whole numbers, \( 4x + 5x - 75 = 10x - 370 \).

By following these steps, you simplify your equation to one involving only whole numbers, which are generally easier to manipulate. This makes the process of solving it quicker and more straightforward.

Linear Equations

Linear equations are the backbone of algebra. These equations consist of constants and a single variable, typically represented by letters like \( x \). The aim is to find the value of this variable that makes the equation true. Let's explore the essential steps in working with linear equations.

• Form the Equation: Linear equations are usually given in the form \( Ax + B = C \). In our example, \( x \) represents the price of the video game system.
• Combine Like Terms: When you have like terms on the same side of the equation, add or subtract them to simplify. For instance, we combined \( \frac{2}{5}x \) and \( \frac{1}{2}x \) by bringing them under a common denominator.
• Isolate the Variable: To solve for \( x \), you need to get it on one side of the equation by itself. Move terms involving \( x \) to one side and constant terms to the opposite side. In our case, moving \( 10x \) to the left side helps in isolating \( x \).

This approach transforms the initially complex-looking equation into a more manageable one. Solving linear equations is a skill you’ll use frequently in algebra, as it allows you to find unknown values based on what is given.

Word Problems in Algebra

Algebraic word problems combine language, logic, and algebra to turn real-world situations into mathematical equations. These problems require careful reading and interpretation to form a solvable equation. Here are some tips to decode and tackle them effectively:

• Understand the Problem: Read the problem thoroughly to identify what is being asked. In our scenario, Allison's savings situation is described step by step, leading to finding the total price.
• Translate Words into Equations: Break down sentences into algebraic expressions. For example, "8 dollars less than \( \frac{2}{5} \) the price" translates to \( \frac{2}{5}x - 8 \).
• Organize Information: Write down all known quantities and relationships. In this problem, we know how much she saves each week and how much she's still short.

Word problems often seem daunting despite being based on everyday scenarios. The key is to approach them one step at a time, transforming complex sentence structures into neat linear equations you’re already familiar with. This structured way of dealing with word problems bridges the gap between theoretical math and practical application.