What Is The Solution To The Linear Equation?$ -12 + 3b - 1 = -5 - B $A. $ B = -2 $ B. $ B = -1.5 $ C. $ B = 1.5 $ D. $ B = 2 $

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Introduction

In mathematics, a linear equation is an equation in which the highest power of the variable(s) is 1. It is a fundamental concept in algebra and is used to solve for the value of the variable(s) in a given equation. In this article, we will explore the solution to a linear equation, specifically the equation −12+3b−1=−5−b-12 + 3b - 1 = -5 - b.

Understanding the Equation

The given equation is −12+3b−1=−5−b-12 + 3b - 1 = -5 - b. To solve for the value of bb, we need to isolate the variable bb on one side of the equation. The first step is to simplify the equation by combining like terms.

Simplifying the Equation

To simplify the equation, we need to combine the constant terms on the left-hand side of the equation.

−12+3b−1=−5−b-12 + 3b - 1 = -5 - b

Combine the constant terms:

−13+3b=−5−b-13 + 3b = -5 - b

Isolating the Variable

Now that we have simplified the equation, we need to isolate the variable bb on one side of the equation. We can do this by adding bb to both sides of the equation.

−13+3b+b=−5−b+b-13 + 3b + b = -5 - b + b

Combine like terms:

−13+4b=−5-13 + 4b = -5

Solving for b

Now that we have isolated the variable bb, we can solve for its value. We can do this by adding 13 to both sides of the equation.

−13+4b+13=−5+13-13 + 4b + 13 = -5 + 13

Combine like terms:

4b=84b = 8

Dividing Both Sides

To solve for the value of bb, we need to divide both sides of the equation by 4.

4b4=84\frac{4b}{4} = \frac{8}{4}

Simplify:

b=2b = 2

Conclusion

In conclusion, the solution to the linear equation −12+3b−1=−5−b-12 + 3b - 1 = -5 - b is b=2b = 2. This means that the value of bb that satisfies the equation is 2.

Answer

The correct answer is:

A. b=−2b = -2

B. b=−1.5b = -1.5

C. b=1.5b = 1.5

D. b=2b = 2

The correct answer is D. b=2b = 2.

Why is this Important?

Understanding how to solve linear equations is an essential skill in mathematics and is used in a wide range of applications, including physics, engineering, and economics. It is also a fundamental concept in algebra and is used to solve more complex equations.

Real-World Applications

Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under the influence of forces.
  • Engineering: Linear equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Linear equations are used to model economic systems and make predictions about economic trends.

Tips and Tricks

Here are some tips and tricks for solving linear equations:

  • Always simplify the equation by combining like terms.
  • Isolate the variable on one side of the equation.
  • Use inverse operations to solve for the value of the variable.
  • Check your answer by plugging it back into the original equation.

Conclusion

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 is a fundamental concept in algebra and is used to solve for the value of the variable(s) in a given equation.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you need to combine like terms on both sides of the equation. This involves adding or subtracting the coefficients of the same variables.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, you need to use inverse operations to get the variable by itself on one side of the equation. This involves adding or subtracting the same value to both sides of the equation.

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 linear equation with fractions?

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

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

A: Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your answer by plugging it 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 simplifying the equation before solving
  • Not isolating the variable on one side of the equation
  • Not checking the answer by plugging it back into the original equation

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

A: To check your answer when solving a linear equation, you need to plug the solution back into the original equation and verify that it is true.

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 wide range of fields, including physics, engineering, and economics.

Q: What are some examples of real-world problems that can be solved using linear equations?

A: Some examples of real-world problems that can be solved using linear equations include:

  • Modeling the motion of an object under the influence of forces
  • Designing and optimizing systems, such as bridges and buildings
  • Making predictions about economic trends

Q: How do I apply linear equations to solve real-world problems?

A: To apply linear equations to solve real-world problems, you need to:

  • Identify the variables and constants in the problem
  • Write an equation that represents the problem
  • Solve the equation using algebraic techniques
  • Interpret the solution in the context of the problem

Conclusion

In conclusion, linear equations are a fundamental concept in algebra and have many real-world applications. By understanding how to solve linear equations, you can apply this knowledge to a wide range of fields and make predictions about complex systems.