The Solution Is $n=-2$, Verified As A Solution To The Equation $1.4n + 2 = 2n + 3.2$. What Is The Last Line Of The Justification?A. \$0.8 = 0.8$[/tex\]B. $-0.8 = -0.8$C. $3 = 3$D. \$-3

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Introduction

In mathematics, solving linear equations is a fundamental concept that forms the basis of various mathematical operations. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on solving a linear equation of the form 1.4n+2=2n+3.21.4n + 2 = 2n + 3.2. We will use the given solution, n=−2n = -2, to verify its validity and provide a step-by-step justification.

The Given Equation

The given equation is:

1.4n+2=2n+3.21.4n + 2 = 2n + 3.2

Step 1: Verify the Solution

To verify the solution, we need to substitute n=−2n = -2 into the equation and check if it satisfies the equation.

1.4(−2)+2=2(−2)+3.21.4(-2) + 2 = 2(-2) + 3.2

Step 2: Simplify the Equation

Now, let's simplify the equation by performing the arithmetic operations.

−2.8+2=−4+3.2-2.8 + 2 = -4 + 3.2

Step 3: Combine Like Terms

Next, we will combine like terms on both sides of the equation.

−0.8=−0.8-0.8 = -0.8

Conclusion

The last line of the justification is:

−0.8=−0.8-0.8 = -0.8

This confirms that the solution n=−2n = -2 is indeed a valid solution to the equation 1.4n+2=2n+3.21.4n + 2 = 2n + 3.2.

Answer

The correct answer is:

A. $0.8 = 0.8$

However, this is not the correct answer. The correct answer is:

B. $-0.8 = -0.8$

Discussion

The given equation is a linear equation in one variable, nn. The solution n=−2n = -2 is verified by substituting it into the equation and simplifying the resulting expression. The final step confirms that the solution satisfies the equation, making it a valid solution.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS):

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

By following these steps and simplifying the equation, you can verify the solution and confirm its validity.

Common Mistakes

When solving linear equations, some common mistakes to avoid include:

  • Not following the order of operations (PEMDAS)
  • Not simplifying the equation properly
  • Not combining like terms correctly
  • Not verifying the solution by substituting it into the equation

By being aware of these common mistakes, you can avoid them and ensure that your solutions are accurate and valid.

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 can be written in the form:

ax+b=cx+dax + b = cx + d

where aa, bb, cc, and dd are constants, and xx is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

  1. Simplify the equation by combining like terms.
  2. Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
  3. Check your solution by substituting it back into the original equation.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when simplifying an expression. The acronym PEMDAS stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate 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 verify a solution to a linear equation?

A: To verify a solution to a linear equation, substitute the solution back into the original equation and simplify the resulting expression. If the expression is true, then the solution is valid.

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 simplifying the equation properly
  • Not combining like terms correctly
  • Not verifying the solution by substituting it into the equation

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

A: Yes, you can use a calculator to solve linear equations. However, it's essential to understand the steps involved in solving the equation and to verify the solution by substituting it back into the original equation.

Q: How do I graph a linear equation?

A: To graph a linear equation, follow these steps:

  1. Find the x-intercept by setting y = 0 and solving for x.
  2. Find the y-intercept by setting x = 0 and solving for y.
  3. Plot the x-intercept and y-intercept on a coordinate plane.
  4. Draw a line through the two points to represent the linear 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. For example:

  • Linear equation: 2x+3=52x + 3 = 5
  • Quadratic equation: x2+4x+4=0x^2 + 4x + 4 = 0

Q: Can I solve a system of linear equations?

A: Yes, you can solve a system of linear equations by using methods such as substitution, elimination, or graphing.