Understanding the Property Used

Oct 13, 2025 by recoveryshake.com 112 views

Introduction

In mathematics, solving equations is a fundamental concept that involves using various properties of equality to isolate the variable. One of the most common properties used in solving equations is the addition property of equality, which states that if two expressions are equal, then adding the same value to both sides of the equation will result in two new expressions that are also equal. However, in the given equation, the property used is not the addition property of equality. Instead, it involves the use of a different property to solve for the variable.

The Given Equation

The given equation is:

\begin{array}{r} -5x = 4 \\ \frac{5x}{-5} = \frac{4}{-5} \\ x = -\frac{4}{5} \end{array}

Understanding the Property Used

To solve for the variable x, the given equation involves the use of the multiplication property of equality. This property states that if two expressions are equal, then multiplying both sides of the equation by the same non-zero value will result in two new expressions that are also equal.

In the given equation, the first step involves multiplying both sides of the equation by -5, which is a non-zero value. This results in the equation:

\frac{5x}{-5} = \frac{4}{-5}

The second step involves multiplying both sides of the equation by -5 again, which results in the equation:

x = -\frac{4}{5}

Why the Multiplication Property of Equality was Used

The multiplication property of equality was used to solve for the variable x because it allowed us to eliminate the coefficient of x, which was -5. By multiplying both sides of the equation by -5, we were able to get rid of the coefficient and isolate the variable x.

Why Not the Addition Property of Equality?

The addition property of equality was not used to solve for the variable x because it would not have resulted in the correct solution. If we had added 5 to both sides of the equation, we would have gotten:

-5x + 5 = 4 + 5

This would have resulted in a new equation that is not equal to the original equation, and would not have led to the correct solution.

Conclusion

In conclusion, the property of equality used to solve the given equation was the multiplication property of equality. This property was used to eliminate the coefficient of x and isolate the variable. The addition property of equality was not used because it would not have resulted in the correct solution.

Properties of Equality

There are several properties of equality that are used to solve equations, including:

Addition property of equality: If two expressions are equal, then adding the same value to both sides of the equation will result in two new expressions that are also equal.
Subtraction property of equality: If two expressions are equal, then subtracting the same value from both sides of the equation will result in two new expressions that are also equal.
Multiplication property of equality: If two expressions are equal, then multiplying both sides of the equation by the same non-zero value will result in two new expressions that are also equal.
Division property of equality: If two expressions are equal, then dividing both sides of the equation by the same non-zero value will result in two new expressions that are also equal.

Examples of Using the Properties of Equality

Here are some examples of using the properties of equality to solve equations:

Example 1: Solve the equation 2x + 3 = 5 using the addition property of equality.
$2x + 3 = 5$
Subtract 3 from both sides of the equation:
$2x = 5 - 3$
Simplify the right-hand side of the equation:
$2x = 2$
Divide both sides of the equation by 2:
$x = \frac{2}{2}$
Simplify the right-hand side of the equation:
$x = 1$
Example 2: Solve the equation 3x - 2 = 7 using the subtraction property of equality.
$3x - 2 = 7$
Add 2 to both sides of the equation:
$3x = 7 + 2$
Simplify the right-hand side of the equation:
$3x = 9$
Divide both sides of the equation by 3:
$x = \frac{9}{3}$
Simplify the right-hand side of the equation:
$x = 3$
Example 3: Solve the equation 4x = 12 using the multiplication property of equality.
$4x = 12$
Divide both sides of the equation by 4:
$x = \frac{12}{4}$
Simplify the right-hand side of the equation:
$x = 3$
Example 4: Solve the equation x/2 = 3 using the division property of equality.
$\frac{x}{2} = 3$
Multiply both sides of the equation by 2:
$x = 3 \times 2$
Simplify the right-hand side of the equation:
$x = 6$

Conclusion

In conclusion, the properties of equality are essential tools for solving equations. By understanding and applying these properties, we can isolate the variable and solve for its value. The addition property of equality, subtraction property of equality, multiplication property of equality, and division property of equality are all important properties that are used to solve equations.

Final Answer

The final answer is:

A. Multiplication property of equality
Q&A: Properties of Equality =============================

Q: What is the addition property of equality?

A: The addition property of equality states that if two expressions are equal, then adding the same value to both sides of the equation will result in two new expressions that are also equal.

Q: How is the addition property of equality used to solve equations?

A: The addition property of equality is used to solve equations by adding the same value to both sides of the equation. This allows us to eliminate a constant term or to isolate the variable.

Q: What is the subtraction property of equality?

A: The subtraction property of equality states that if two expressions are equal, then subtracting the same value from both sides of the equation will result in two new expressions that are also equal.

Q: How is the subtraction property of equality used to solve equations?

A: The subtraction property of equality is used to solve equations by subtracting the same value from both sides of the equation. This allows us to eliminate a constant term or to isolate the variable.

Q: What is the multiplication property of equality?

A: The multiplication property of equality states that if two expressions are equal, then multiplying both sides of the equation by the same non-zero value will result in two new expressions that are also equal.

Q: How is the multiplication property of equality used to solve equations?

A: The multiplication property of equality is used to solve equations by multiplying both sides of the equation by a non-zero value. This allows us to eliminate a coefficient or to isolate the variable.

Q: What is the division property of equality?

A: The division property of equality states that if two expressions are equal, then dividing both sides of the equation by the same non-zero value will result in two new expressions that are also equal.

Q: How is the division property of equality used to solve equations?

A: The division property of equality is used to solve equations by dividing both sides of the equation by a non-zero value. This allows us to eliminate a coefficient or to isolate the variable.

Q: What are some examples of using the properties of equality to solve equations?

A: Here are some examples of using the properties of equality to solve equations:

Example 1: Solve the equation 2x + 3 = 5 using the addition property of equality.
$2x + 3 = 5$
Subtract 3 from both sides of the equation:
$2x = 5 - 3$
Simplify the right-hand side of the equation:
$2x = 2$
Divide both sides of the equation by 2:
$x = \frac{2}{2}$
Simplify the right-hand side of the equation:
$x = 1$
Example 2: Solve the equation 3x - 2 = 7 using the subtraction property of equality.
$3x - 2 = 7$
Add 2 to both sides of the equation:
$3x = 7 + 2$
Simplify the right-hand side of the equation:
$3x = 9$
Divide both sides of the equation by 3:
$x = \frac{9}{3}$
Simplify the right-hand side of the equation:
$x = 3$
Example 3: Solve the equation 4x = 12 using the multiplication property of equality.
$4x = 12$
Divide both sides of the equation by 4:
$x = \frac{12}{4}$
Simplify the right-hand side of the equation:
$x = 3$
Example 4: Solve the equation x/2 = 3 using the division property of equality.
$\frac{x}{2} = 3$
Multiply both sides of the equation by 2:
$x = 3 \times 2$
Simplify the right-hand side of the equation:
$x = 6$

Conclusion

Final Answer

The final answer is:

A. Multiplication property of equality