Solve The Equation:${ 2.5x - X = 4x }$

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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 focus on solving a specific type of linear equation, which is the equation 2.5x−x=4x2.5x - x = 4x. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.

Understanding Linear Equations

A linear equation is an equation in which the highest power of the variable (in this case, xx) is 1. Linear equations can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants. The goal of solving a linear equation is to isolate the variable, which means to get it by itself on one side of the equation.

Solving the Equation 2.5x−x=4x2.5x - x = 4x

To solve the equation 2.5x−x=4x2.5x - x = 4x, we will follow the order of operations (PEMDAS):

  1. Combine like terms: The first step is to combine the like terms on the left-hand side of the equation. In this case, we have 2.5x2.5x and −x-x, which are like terms because they both contain the variable xx. To combine them, we add their coefficients, which are the numbers in front of the variable. So, 2.5x−x=(2.5−1)x=1.5x2.5x - x = (2.5 - 1)x = 1.5x.
  2. Simplify the equation: Now that we have combined the like terms, we can simplify the equation by writing it as 1.5x=4x1.5x = 4x.
  3. Subtract 4x4x from both sides: To isolate the variable, we need to get rid of the 4x4x on the right-hand side of the equation. We can do this by subtracting 4x4x from both sides of the equation. This gives us 1.5x−4x=01.5x - 4x = 0.
  4. Combine like terms again: Now that we have subtracted 4x4x from both sides, we can combine the like terms on the left-hand side of the equation. In this case, we have 1.5x1.5x and −4x-4x, which are like terms because they both contain the variable xx. To combine them, we add their coefficients, which are the numbers in front of the variable. So, 1.5x−4x=(−4+1.5)x=−2.5x1.5x - 4x = (-4 + 1.5)x = -2.5x.
  5. Solve for xx: Now that we have combined the like terms, we can solve for xx by dividing both sides of the equation by −2.5-2.5. This gives us x=0x = 0.

Conclusion

In this article, we solved the equation 2.5x−x=4x2.5x - x = 4x by following the order of operations (PEMDAS). We combined like terms, simplified the equation, subtracted 4x4x from both sides, combined like terms again, and finally solved for xx. The solution to the equation is x=0x = 0.

Tips and Tricks

  • When solving linear equations, always follow the order of operations (PEMDAS).
  • Combine like terms whenever possible to simplify the equation.
  • Subtract or add the same value to both sides of the equation to isolate the variable.
  • Divide both sides of the equation by the coefficient of the variable to solve for xx.

Real-World Applications

Linear equations have many real-world applications, including:

  • Finance: Linear equations are used to calculate interest rates, investment returns, and loan payments.
  • Science: Linear equations are used to model population growth, chemical reactions, and physical systems.
  • Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Practice Problems

Try solving the following linear equations:

  • 3x+2=5x3x + 2 = 5x
  • 2x−3=x+12x - 3 = x + 1
  • 4x+2=3x−14x + 2 = 3x - 1

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following the order of operations (PEMDAS) and combining like terms, we can simplify the equation and solve for the variable. Linear equations have many real-world applications, including finance, science, and engineering. With practice and patience, anyone can become proficient in solving linear equations.

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Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, xx) is 1. Linear equations can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow the order of operations (PEMDAS):

  1. Combine like terms on the left-hand side of the equation.
  2. Simplify the equation by writing it in the form ax=bax = b.
  3. Divide both sides of the equation by aa to solve for xx.

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 (in this case, xx) is 1. A quadratic equation is an equation in which the highest power of the variable (in this case, xx) is 2. For example, 2x+3=52x + 3 = 5 is a linear equation, while x2+4x+4=0x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I know if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, look at the highest power of the variable (in this case, xx). If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your work by plugging the solution back into the original equation.

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

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

  • Not following the order of operations (PEMDAS)
  • Not combining like terms
  • Not simplifying the equation
  • Not checking your work by plugging the solution back into the original equation

Q: Can I use linear equations to solve real-world problems?

A: Yes, linear equations can be used to solve real-world problems in a variety of fields, including finance, science, and engineering.

Q: How do I know if a linear equation has a solution?

A: A linear equation has a solution if the equation is consistent, meaning that the left-hand side and right-hand side of the equation are equal. If the equation is inconsistent, meaning that the left-hand side and right-hand side of the equation are not equal, then the equation has no solution.

Q: Can I use linear equations to solve systems of equations?

A: Yes, linear equations can be used to solve systems of equations. A system of equations is a set of two or more linear equations that are solved simultaneously.

Q: How do I know if a system of linear equations has a solution?

A: A system of linear equations has a solution if the equations are consistent, meaning that the left-hand side and right-hand side of each equation are equal. If the equations are inconsistent, meaning that the left-hand side and right-hand side of each equation are not equal, then the system has no solution.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following the order of operations (PEMDAS) and combining like terms, we can simplify the equation and solve for the variable. Linear equations have many real-world applications, including finance, science, and engineering. With practice and patience, anyone can become proficient in solving linear equations.