Ashley Solved The Equation For $m$ Using The Following Steps:$\[ \begin{array}{l} -4.5 + M = 17 \\ \quad \quad \quad \quad -4.5 \\ \underline{\qquad M = 12.5} \end{array} \\]What Was Ashley's Error?A. Ashley Should Have Added 4.5 To

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Introduction

Solving equations is a fundamental concept in mathematics, and it requires careful attention to detail to arrive at the correct solution. In this article, we will analyze the steps taken by Ashley to solve the equation for $m$ and identify the error in her approach.

The Equation and Ashley's Steps

The equation given to Ashley is:

$$-4.5 + m = 17$$

Ashley's steps to solve the equation are as follows:

$$\begin{array}{l} -4.5 + m = 17 \\ \quad \quad \quad \quad -4.5 \\ \underline{\qquad m = 12.5} \end{array}$$

Analyzing Ashley's Error

At first glance, it appears that Ashley has correctly solved the equation. However, upon closer inspection, we can see that there is an error in her approach. The error lies in the fact that Ashley has added $-4.5$ to both sides of the equation, but she has not added $4.5$ to the right-hand side of the equation.

Correcting the Error

To correct the error, Ashley should have added $4.5$ to both sides of the equation. This would have resulted in the following steps:

$$\begin{array}{l} -4.5 + m = 17 \\ \quad \quad \quad \quad +4.5 \\ \underline{\qquad m = 21.5} \end{array}$$

Conclusion

In conclusion, Ashley's error in solving the equation for $m$ was that she did not add $4.5$ to the right-hand side of the equation. By adding $4.5$ to both sides of the equation, Ashley would have arrived at the correct solution, which is $m = 21.5$.

Common Mistakes in Solving Equations

There are several common mistakes that students make when solving equations. Some of these mistakes include:

• Not adding or subtracting the same value to both sides of the equation
• Not multiplying or dividing both sides of the equation by the same value
• Not checking the solution to ensure that it satisfies the original equation

Tips for Solving Equations

To avoid making mistakes when solving equations, follow these tips:

• Read the equation carefully and understand what is being asked
• Check the solution to ensure that it satisfies the original equation
• Use inverse operations to isolate the variable
• Check your work to ensure that you have not made any mistakes

Real-World Applications of Solving Equations

Solving equations has numerous real-world applications. Some of these applications include:

• Physics and engineering: Solving equations is used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
• Computer science: Solving equations is used in computer algorithms, such as linear programming and optimization.
• Economics: Solving equations is used to model economic systems and make predictions about future economic trends.

Conclusion

Q: What is an equation?

A: An equation is a statement that two mathematical expressions are equal. It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), which are separated by an equal sign (=).

Q: What is the purpose of solving an equation?

A: The purpose of solving an equation is to find the value of the variable (usually represented by a letter such as x or y) that makes the equation true.

Q: What are the steps to solve an equation?

A: The steps to solve an equation are:

1. Read the equation carefully: Understand what is being asked and what the equation is saying.
2. Check for any parentheses: Evaluate any expressions inside parentheses first.
3. Use inverse operations: Use the opposite operation to isolate the variable.
4. Add or subtract the same value to both sides: Add or subtract the same value to both sides of the equation to isolate the variable.
5. Multiply or divide both sides by the same value: Multiply or divide both sides of the equation by the same value to isolate the variable.
6. Check the solution: Check the solution to ensure that it satisfies the original 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 is 1. For example, 2x + 3 = 5 is a linear equation. A quadratic equation is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: What is the difference between a system of equations and a single equation?

A: A system of equations is a set of two or more equations that are solved simultaneously. A single equation is a single equation that is solved independently.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use the following methods:

1. Substitution method: Substitute the expression for one variable from one equation into the other equation.
2. Elimination method: Add or subtract the equations to eliminate one variable.
3. Graphing method: Graph the equations on a coordinate plane and find the point of intersection.

Q: What is the importance of solving equations in real-life situations?

A: Solving equations is important in real-life situations because it helps us:

1. Model real-world phenomena: Equations can be used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
2. Make predictions: Equations can be used to make predictions about future events, such as the price of a stock or the weather.
3. Optimize systems: Equations can be used to optimize systems, such as finding the maximum or minimum value of a function.

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

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

1. Not adding or subtracting the same value to both sides of the equation
2. Not multiplying or dividing both sides of the equation by the same value
3. Not checking the solution to ensure that it satisfies the original equation

Conclusion

In conclusion, solving equations is a fundamental concept in mathematics that has numerous real-world applications. By understanding the steps to solve an equation, the difference between a linear equation and a quadratic equation, and the importance of solving equations in real-life situations, you can avoid making mistakes and arrive at the correct solution.