Gamal Spent $\$12.50$ At The Bookstore. The Difference Between The Amount He Spent At The Video Game Store And The Amount He Spent At The Bookstore Was $\$17$. The Equation $d - 12.50 = 17$ Can Be Used To Represent This

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Introduction

Linear equations are a fundamental concept in mathematics, and they have numerous real-world applications. In this article, we will explore how linear equations can be used to represent real-world scenarios, using the example of Gamal's shopping trip to the bookstore and the video game store.

The Problem

Gamal spent $12.50\$12.50 at the bookstore. The difference between the amount he spent at the video game store and the amount he spent at the bookstore was $17\$17. We can use this information to set up a linear equation to represent the situation.

Setting Up the Equation

Let's denote the amount Gamal spent at the video game store as dd. We know that the difference between the amount he spent at the video game store and the amount he spent at the bookstore was $17\$17. This can be represented by the equation:

d−12.50=17d - 12.50 = 17

Solving the Equation

To solve for dd, we need to isolate the variable on one side of the equation. We can do this by adding 12.5012.50 to both sides of the equation:

d−12.50+12.50=17+12.50d - 12.50 + 12.50 = 17 + 12.50

This simplifies to:

d=29.50d = 29.50

Interpretation

So, Gamal spent $29.50\$29.50 at the video game store. This makes sense, since the difference between the amount he spent at the video game store and the amount he spent at the bookstore was $17\$17, and he spent $12.50\$12.50 at the bookstore.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Finance: Linear equations can be used to represent the balance of a bank account, the cost of goods sold, and the profit or loss of a business.
  • Science: Linear equations can be used to represent the motion of objects, the growth of populations, and the behavior of physical systems.
  • Engineering: Linear equations can be used to represent the design of electrical circuits, the behavior of mechanical systems, and the performance of computer algorithms.

Conclusion

In this article, we have seen how linear equations can be used to represent real-world scenarios, using the example of Gamal's shopping trip to the bookstore and the video game store. We have also explored some of the real-world applications of linear equations, including finance, science, and engineering.

Tips for Solving Linear Equations

  • Isolate the variable: To solve for the variable, isolate it on one side of the equation.
  • Use inverse operations: Use inverse operations, such as addition and subtraction, to isolate the variable.
  • Check your work: Check your work by plugging the solution back into the original equation.

Practice Problems

  1. Solve the equation x+5=11x + 5 = 11 for xx.
  2. Solve the equation 2y−3=72y - 3 = 7 for yy.
  3. Solve the equation z−2=9z - 2 = 9 for zz.

Answer Key

  1. x=6x = 6
  2. y=5y = 5
  3. z=11z = 11

References

  • Algebra: A Comprehensive Introduction by Michael Artin
  • Linear Algebra: A Modern Introduction by David C. Lay
  • Mathematics for Computer Science: A Modern Introduction by Eric Lehman and Tom Leighton
    Frequently Asked Questions (FAQs) About Solving Linear Equations ====================================================================

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form of ax + b = c, where a, b, and c are constants, and x is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by using inverse operations, such as addition and subtraction, to get the variable by itself.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

  • Not isolating the variable: Make sure to isolate the variable on one side of the equation.
  • Not using inverse operations: Use inverse operations, such as addition and subtraction, to get the variable by itself.
  • Not checking your work: Check your work by plugging the solution back into the original equation.

Q: How do I check my work when solving a linear equation?

A: To check your work, plug the solution back into the original equation and make sure it is true. If it is not true, then you made a mistake and need to go back and try again.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications, including:

  • Finance: Linear equations can be used to represent the balance of a bank account, the cost of goods sold, and the profit or loss of a business.
  • Science: Linear equations can be used to represent the motion of objects, the growth of populations, and the behavior of physical systems.
  • Engineering: Linear equations can be used to represent the design of electrical circuits, the behavior of mechanical systems, and the performance of computer algorithms.

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, you need to get rid of the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: How do I solve a linear equation with decimals?

A: To solve a linear equation with decimals, you need to get rid of the decimals by multiplying both sides of the equation by a power of 10.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a.

Q: What are some common types of linear equations?

A: Some common types of linear equations include:

  • Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
  • Standard form: ax + by = c, where a, b, and c are constants.
  • Point-slope form: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find two points on the line and plot them on a coordinate plane. Then, draw a line through the two points to represent the equation.

Q: What are some common mistakes to avoid when graphing a linear equation?

A: Some common mistakes to avoid when graphing a linear equation include:

  • Not finding two points on the line: Make sure to find two points on the line and plot them on a coordinate plane.
  • Not drawing a line through the two points: Draw a line through the two points to represent the equation.
  • Not labeling the axes: Label the x-axis and y-axis to make it easier to read the graph.