Examine The Steps Used To Solve The Equation:${12.5x - 10.2 = 3(2.5x + 4.2) - 6}$1. ${12.5x - 10.2 = 7.5x + 12.6 - 6}$2. ${12.5x - 10.2 = 7.5x + 6.6}$3. ${12.5x = 7.5x + 16.8}$4. ${5x = 16.8}$5.

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will examine the steps used to solve a linear equation, using the equation 12.5x−10.2=3(2.5x+4.2)−612.5x - 10.2 = 3(2.5x + 4.2) - 6 as an example.

Step 1: Distribute the Coefficient

The first step in solving the equation is to distribute the coefficient of the term inside the parentheses. In this case, we have 3(2.5x+4.2)3(2.5x + 4.2), which can be expanded as 7.5x+12.67.5x + 12.6.

# Distribute the coefficient
equation = "12.5x - 10.2 = 3(2.5x + 4.2) - 6"
distributed_equation = "12.5x - 10.2 = 7.5x + 12.6 - 6"
print(distributed_equation)

Step 2: Combine Like Terms

The next step is to combine like terms on both sides of the equation. In this case, we can combine the constant terms on the right-hand side of the equation.

# Combine like terms
distributed_equation = "12.5x - 10.2 = 7.5x + 12.6 - 6"
combined_equation = "12.5x - 10.2 = 7.5x + 6.6"
print(combined_equation)

Step 3: Isolate the Variable

The next step is to isolate the variable xx on one side of the equation. In this case, we can do this by subtracting 7.5x7.5x from both sides of the equation.

# Isolate the variable
combined_equation = "12.5x - 10.2 = 7.5x + 6.6"
isolated_equation = "12.5x - 7.5x - 10.2 = 6.6"
print(isolated_equation)

Step 4: Simplify the Equation

The next step is to simplify the equation by combining like terms.

# Simplify the equation
isolated_equation = "12.5x - 7.5x - 10.2 = 6.6"
simplified_equation = "5x - 10.2 = 6.6"
print(simplified_equation)

Step 5: Add the Constant Term

The next step is to add the constant term to both sides of the equation.

# Add the constant term
simplified_equation = "5x - 10.2 = 6.6"
added_equation = "5x = 6.6 + 10.2"
print(added_equation)

Step 6: Simplify the Right-Hand Side

The next step is to simplify the right-hand side of the equation.

# Simplify the right-hand side
added_equation = "5x = 6.6 + 10.2"
simplified_rhs = "5x = 16.8"
print(simplified_rhs)

Step 7: Solve for the Variable

The final step is to solve for the variable xx by dividing both sides of the equation by the coefficient of the variable.

# Solve for the variable
simplified_rhs = "5x = 16.8"
solution = "x = 16.8 / 5"
print(solution)

Conclusion

In this article, we examined the steps used to solve a linear equation, using the equation 12.5x−10.2=3(2.5x+4.2)−612.5x - 10.2 = 3(2.5x + 4.2) - 6 as an example. We distributed the coefficient, combined like terms, isolated the variable, simplified the equation, added the constant term, simplified the right-hand side, and finally solved for the variable. By following these steps, we can solve linear equations with ease.

Discussion

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we examined the steps used to solve a linear equation, using the equation 12.5x−10.2=3(2.5x+4.2)−612.5x - 10.2 = 3(2.5x + 4.2) - 6 as an example. We distributed the coefficient, combined like terms, isolated the variable, simplified the equation, added the constant term, simplified the right-hand side, and finally solved for the variable.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future economic trends.
  • Computer Science: Linear equations are used in computer graphics and game development to create realistic simulations.

Conclusion

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will answer some of the most frequently asked questions 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 is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants.

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. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations when solving a linear equation?

A: The order of operations when solving a linear equation is:

  1. Distribute the coefficient
  2. Combine like terms
  3. Isolate the variable
  4. Simplify the equation
  5. Add the constant term
  6. Simplify the right-hand side
  7. Solve for the variable

Q: How do I distribute the coefficient?

A: To distribute the coefficient, you need to multiply the coefficient by each term inside the parentheses.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms.

Q: How do I isolate the variable?

A: To isolate the variable, you need to add, subtract, multiply, or divide both sides of the equation by the same value.

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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a quadratic equation?

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

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula and simplify.

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

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

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future economic trends.
  • Computer Science: Linear equations are used in computer graphics and game development to create realistic simulations.

Conclusion

In conclusion, linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By following the steps outlined in this article, we can solve linear equations with ease. Linear equations have numerous real-world applications, including physics, engineering, economics, and computer science.

Frequently Asked Questions

  • Q: What is a linear equation? A: A linear equation is an equation in which the highest power of the variable is 1.
  • 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.
  • Q: What is the order of operations when solving a linear equation? A: The order of operations when solving a linear equation is: 1. Distribute the coefficient 2. Combine like terms 3. Isolate the variable 4. Simplify the equation 5. Add the constant term 6. Simplify the right-hand side 7. Solve for the variable
  • Q: How do I distribute the coefficient? A: To distribute the coefficient, you need to multiply the coefficient by each term inside the parentheses.
  • Q: How do I combine like terms? A: To combine like terms, you need to add or subtract the coefficients of the like terms.
  • Q: How do I isolate the variable? A: To isolate the variable, you need to add, subtract, multiply, or divide both sides of the equation by the same value.

Additional Resources

  • Linear Equation Calculator: A calculator that can be used to solve linear equations.
  • Linear Equation Solver: A solver that can be used to solve linear equations.
  • Linear Equation Tutorial: A tutorial that provides a step-by-step guide to solving linear equations.