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Chapter 12: Problem 13

Use the following information to answer the next three questions. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is \(y=12,000 x\) . How many pounds of soil does the shoreline lose in a year?

Short Answer

Expert verified
12,000 pounds of soil per year.

Step by step solution

01

Identify the Variables

In the equation given, \(y = 12,000x\), \(y\) represents the total amount of soil lost in pounds, and \(x\) represents the number of years.
02

Interpret the Coefficient

The coefficient of \(x\) in the equation is 12,000. This indicates that each year, the shoreline loses 12,000 pounds of soil.
03

Solve for One Year

To find out how much soil is lost in one year, substitute \(x = 1\) into the equation: \(y = 12,000 \times 1\).
04

Calculate the Total Loss

Perform the multiplication: \(y = 12,000 \times 1 = 12,000\).
05

Conclusion

The shoreline loses 12,000 pounds of soil per year.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Variables in Linear Equations
In the world of linear equations, variables play a crucial role. They are symbols used to represent numbers within an equation. In the linear equation given as \( y = 12,000x \), the symbols \( y \) and \( x \) are the variables.

In this context:
  • \( y \) represents the total amount of soil lost, measured in pounds.
  • \( x \) indicates the number of years, a measure of time over which the soil is lost.


Variables allow equations to be adjusted and solved for different values, providing flexibility. By assigning different numbers to \( x \), we can explore various scenarios for soil loss over time. This concept is essential in any algebra course and helps students develop problem-solving skills.
Interpreting Coefficients in Linear Equations
A coefficient is a number that stands in front of a variable in an equation. It interprets how changes in the variable affect the equation's outcome. In \( y = 12,000x \), 12,000 is the coefficient associated with \( x \).

This coefficient has a real-world implication. It shows that the shoreline loses 12,000 pounds of soil for every year that passes.
  • Multiplying this coefficient by the number of years shows overall soil loss over any given period.
  • In simpler terms, it indicates a linear relationship where soil loss increases steadily with time.


Interpreting coefficients is key to understanding how different factors contribute to changes in an equation's output. This skill is handy, not just in mathematics, but in analyzing real-world problems.
Effective Problem-Solving Steps
Solving linear equations involves a systematic approach. A clear set of steps can enhance problem-solving accuracy and efficiency. Here is a simplified breakdown:

  • Identify the Variables: Determine what each variable in the equation stands for, as we did with \( x \) and \( y \).
  • Interpret the Coefficients: Recognize what the coefficient represents in the real-world scenario, like the annual soil loss per year here.
  • Substitute Values: Use known values for the variables to find unknown ones. For instance, replacing \( x \) with 1 in the equation \( y = 12,000 \cdot 1 \) helped determine the soil loss for one year.
  • Calculate the Result: Performing the mathematical operations, in this case multiplying 12,000 by 1, provides the final answer.
  • Conclude: Summarize the result clearly; as seen, the shoreline loses 12,000 pounds of soil each year.


These steps can be adapted to tackle various linear equation problems, fostering analytical thinking and boosting mathematical proficiency.

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