Problem 55 - Algebra 1: Concepts and Skills

Question

In Exercises 54–56, match the real-life problem with an equation. Then solve the problem. A. \(x+15=7\) B. \(15-x=7\) C. \(15+7=x\) D. \(x+15=-7\) There are 15 members of a high school band brass section. After graduation there are only 7 members. How many members graduated?

Step-by-Step Solution

Answer

There were 8 members who graduated from the high school band's brass section.

Step 1: Analyze the real-life problem

The band originally consists of 15 members, but after graduation, there are only 7 members. So the amount of members who graduated must be such that their subtraction from the initial total (15) equals the number left after graduation (7).

Step 2: Match the real-life problem with an equation

Looking at the given equations, the one that translates this scenario best is Equation B: \(15-x=7\). This equation implies that there were 15 members initially, and when x number of members graduate, the total is reduced to 7. This matches our problem in which x is the number of members who graduated.

Step 3: Solve the equation

To solve the equation \(15-x=7\), we need to isolate x. To do so, we can add x to both sides to get rid of it on the left, then subtract 7 from both sides to find x. That gives us \[15 - 7 = x\], so \(x = 8\].

Key Concepts

Real-Life Problems

Linear equations often emerge in real-life situations, such as determining how many band members graduated. These equations help us solve problems by representing relationships numerically. In the given example, we know there were originally 15 members, and after graduation, only 7 remain. This scenario prompts you to calculate the number of members who left.
By translating this problem into an equation, you can apply mathematical reasoning to find the answer. In essence, real-life problems can be expressed through equations, enabling us to quantify changes and relationships we observe in day-to-day life.

Subtraction Equations

Subtraction equations are a specific type of linear equation where one quantity is subtracted from another. In the exercise, we have an equation of the form:

• \(15-x=7\)

The equation tells us that some quantity, represented by \(x\), was removed from the original total of 15, leaving 7. Subtraction equations like this one help determine the unknown quantity being removed.
Such equations allow us to understand what happens when a part of a total amount is taken away. Solving subtraction equations is a foundational skill in algebra, making it easier to address related problems in real-life contexts like the one in this exercise.

Equation Solving Steps

To solve subtraction equations efficiently, follow these steps. First, understand the problem context and match it to an equation. For our example:

• Identify the original total value: 15 members.
• Determine the remaining value: 7 members.
• Match this context with the equation: \(15-x=7\).
• Isolate the variable \(x\) by performing reverse operations.

For our specific equation, eliminate \(x\) from the negative side by adding \(x\) to both sides, giving us \(15 = x + 7\).
Then, subtract 7 from both sides to solve for \(x\), leading to \(x = 15 - 7\).
Simplifying this, we find \(x = 8\), indicating that 8 members graduated.
Learning to systematically solve equations through these steps is crucial, enabling the conversion of real-world scenarios into manageable mathematical problems.