Solve The Equation: $\[ 1 - [2x - 3(4x - 5)] = -14 \\]

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Introduction

Algebra is a branch of mathematics that deals with solving equations and manipulating variables to find their values. Equations are statements that express the equality of two mathematical expressions, and solving them is a crucial skill in mathematics and science. In this article, we will focus on solving a specific type of equation, and we will use the equation ${ 1 - [2x - 3(4x - 5)] = -14 }$ as an example.

Understanding the Equation

Before we start solving the equation, let's break it down and understand what it means. The equation is a linear equation, which means it has a single variable (x) and a constant term. The equation is also a combination of several operations, including addition, subtraction, multiplication, and division.

The equation can be rewritten as:

1−2x+3(4x−5)=−14{ 1 - 2x + 3(4x - 5) = -14 }

Step 1: Simplify the Equation

To simplify the equation, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: 3(4x−5)=12x−153(4x - 5) = 12x - 15
  2. Rewrite the equation: ${ 1 - 2x + 12x - 15 = -14 }$
  3. Combine like terms: ${ -15 + 10x = -14 }$

Step 2: Isolate the Variable

To isolate the variable (x), we need to get rid of the constant term (-15) on the left-hand side of the equation. We can do this by adding 15 to both sides of the equation:

−15+15+10x=−14+15{ -15 + 15 + 10x = -14 + 15 }

This simplifies to:

10x=1{ 10x = 1 }

Step 3: Solve for x

To solve for x, we need to divide both sides of the equation by 10:

10x10=110{ \frac{10x}{10} = \frac{1}{10} }

This simplifies to:

x=110{ x = \frac{1}{10} }

Conclusion

Solving equations is a crucial skill in mathematics and science. By following the order of operations and isolating the variable, we can solve even the most complex equations. In this article, we used the equation ${ 1 - [2x - 3(4x - 5)] = -14 }$ as an example and walked through the steps to solve it. We simplified the equation, isolated the variable, and solved for x.

Tips and Tricks

  • Always follow the order of operations (PEMDAS)
  • Isolate the variable by getting rid of the constant term
  • Use algebraic properties, such as the distributive property, to simplify the equation
  • Check your work by plugging the solution back into the original equation

Real-World Applications

Solving equations has many real-world applications, including:

  • Physics: Solving equations is essential in physics to describe the motion of objects and predict their behavior.
  • Engineering: Solving equations is crucial in engineering to design and optimize systems.
  • Economics: Solving equations is used in economics to model economic systems and make predictions about future trends.

Common Mistakes

  • Not following the order of operations (PEMDAS)
  • Not isolating the variable
  • Not checking work by plugging the solution back into the original equation

Conclusion

Introduction

Solving equations is a crucial skill in mathematics and science. In our previous article, we walked through the steps to solve a specific type of equation. In this article, we will answer some common questions about solving equations and provide additional tips and tricks to help you master this skill.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an equation?

A: To simplify an equation, we need to follow the order of operations and combine like terms. Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Q: How do I isolate the variable?

A: To isolate the variable, we need to get rid of the constant term on the left-hand side of the equation. We can do this by adding or subtracting the same value to both sides of the equation.

Q: What is the distributive property?

A: The distributive property is a property of algebra that allows us to distribute a coefficient to multiple terms inside parentheses. For example, 3(2x + 4) can be rewritten as 6x + 12 using the distributive property.

Q: How do I check my work?

A: To check your work, we need to plug the solution back into the original equation and make sure it is true. This is called "substituting" the solution into the equation.

Q: What are some common mistakes to avoid?

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

  • Not following the order of operations (PEMDAS)
  • Not isolating the variable
  • Not checking work by plugging the solution back into the original equation
  • Not using algebraic properties, such as the distributive property, to simplify the equation

Q: How do I solve equations with fractions?

A: To solve equations with fractions, we need to follow the same steps as solving equations with integers. However, we need to be careful when multiplying or dividing fractions to avoid making mistakes.

Q: How do I solve equations with decimals?

A: To solve equations with decimals, we need to follow the same steps as solving equations with integers. However, we need to be careful when multiplying or dividing decimals to avoid making mistakes.

Conclusion

Solving equations is a fundamental skill in mathematics and science. By following the order of operations and isolating the variable, we can solve even the most complex equations. In this article, we answered some common questions about solving equations and provided additional tips and tricks to help you master this skill.

Tips and Tricks

  • Always follow the order of operations (PEMDAS)
  • Isolate the variable by getting rid of the constant term
  • Use algebraic properties, such as the distributive property, to simplify the equation
  • Check your work by plugging the solution back into the original equation

Real-World Applications

Solving equations has many real-world applications, including:

  • Physics: Solving equations is essential in physics to describe the motion of objects and predict their behavior.
  • Engineering: Solving equations is crucial in engineering to design and optimize systems.
  • Economics: Solving equations is used in economics to model economic systems and make predictions about future trends.

Common Mistakes

  • Not following the order of operations (PEMDAS)
  • Not isolating the variable
  • Not checking work by plugging the solution back into the original equation
  • Not using algebraic properties, such as the distributive property, to simplify the equation