Select The Property Of Equality Used To Arrive At The Conclusion.If 1 3 X = 5 \frac{1}{3} X = 5 3 1 ​ X = 5 , Then X = 15 X = 15 X = 15 .A. The Subtraction Property Of EqualityB. The Multiplication Property Of EqualityC. The Addition Property Of Equality

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Introduction

In mathematics, solving linear equations is a fundamental concept that involves manipulating equations to isolate the variable. One of the key properties used in solving linear equations is the property of equality. In this article, we will explore the different properties of equality and how they are used to solve linear equations.

What are the Properties of Equality?

The properties of equality are rules that allow us to manipulate equations without changing their solutions. There are three main properties of equality: the addition property of equality, the subtraction property of equality, and the multiplication property of equality.

The Addition Property of Equality

The addition property of equality states that if we add the same value to both sides of an equation, the equation remains true. Mathematically, this can be represented as:

a = b

a + c = b + c

where a, b, and c are any real numbers.

The Subtraction Property of Equality

The subtraction property of equality states that if we subtract the same value from both sides of an equation, the equation remains true. Mathematically, this can be represented as:

a = b

a - c = b - c

where a, b, and c are any real numbers.

The Multiplication Property of Equality

The multiplication property of equality states that if we multiply both sides of an equation by the same non-zero value, the equation remains true. Mathematically, this can be represented as:

a = b

ac = bc

where a, b, and c are any real numbers.

Solving the Given Equation

Now, let's apply the properties of equality to solve the given equation:

13x=5\frac{1}{3} x = 5

To solve for x, we need to isolate x on one side of the equation. We can do this by multiplying both sides of the equation by 3, which is the reciprocal of 13\frac{1}{3}.

Using the Multiplication Property of Equality

By multiplying both sides of the equation by 3, we get:

13x×3=5×3\frac{1}{3} x \times 3 = 5 \times 3

This simplifies to:

x = 15

Therefore, the solution to the equation is x = 15.

Conclusion

In conclusion, the property of equality used to arrive at the conclusion is the multiplication property of equality. This property allows us to multiply both sides of an equation by the same non-zero value, which in this case is 3. By applying this property, we were able to isolate x and solve for its value.

Answer

The correct answer is:

B. The multiplication property of equality

Why is this the correct answer?

This is the correct answer because the equation 13x=5\frac{1}{3} x = 5 requires us to multiply both sides of the equation by 3 to isolate x. This is a direct application of the multiplication property of equality, which states that if we multiply both sides of an equation by the same non-zero value, the equation remains true.

Real-World Applications

The properties of equality have numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, the law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. This is a direct application of the addition property of equality, which states that if we add the same value to both sides of an equation, the equation remains true.

Final Thoughts

In conclusion, the properties of equality are fundamental concepts in mathematics that allow us to manipulate equations without changing their solutions. By understanding and applying these properties, we can solve linear equations and make sense of the world around us. Whether it's solving a simple equation or modeling complex real-world phenomena, the properties of equality are essential tools that every mathematician and scientist should know.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Linear Algebra and Its Applications" by Gilbert Strang
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

  • Khan Academy: Linear Equations
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram MathWorld: Linear Equations
    Frequently Asked Questions: Properties of Equality =====================================================

Q: What is the addition property of equality?

A: The addition property of equality states that if we add the same value to both sides of an equation, the equation remains true. Mathematically, this can be represented as:

a = b

a + c = b + c

where a, b, and c are any real numbers.

Q: What is the subtraction property of equality?

A: The subtraction property of equality states that if we subtract the same value from both sides of an equation, the equation remains true. Mathematically, this can be represented as:

a = b

a - c = b - c

where a, b, and c are any real numbers.

Q: What is the multiplication property of equality?

A: The multiplication property of equality states that if we multiply both sides of an equation by the same non-zero value, the equation remains true. Mathematically, this can be represented as:

a = b

ac = bc

where a, b, and c are any real numbers.

Q: How do I apply the properties of equality to solve linear equations?

A: To apply the properties of equality to solve linear equations, follow these steps:

  1. Identify the equation you want to solve.
  2. Determine which property of equality to apply (addition, subtraction, or multiplication).
  3. Apply the property to both sides of the equation.
  4. Simplify the equation to isolate the variable.

Q: What are some common mistakes to avoid when applying the properties of equality?

A: Some common mistakes to avoid when applying the properties of equality include:

  • Adding or subtracting the wrong value to both sides of the equation.
  • Multiplying both sides of the equation by a zero value.
  • Not simplifying the equation after applying the property.

Q: How do I know which property of equality to apply?

A: To determine which property of equality to apply, follow these steps:

  1. Look at the equation and identify the variable you want to isolate.
  2. Determine what operation is needed to isolate the variable (addition, subtraction, or multiplication).
  3. Apply the corresponding property of equality to both sides of the equation.

Q: Can I apply the properties of equality to solve quadratic equations?

A: Yes, you can apply the properties of equality to solve quadratic equations. However, you may need to use additional techniques, such as factoring or the quadratic formula, to solve the equation.

Q: Are the properties of equality only used in algebra?

A: No, the properties of equality are used in many areas of mathematics, including geometry, trigonometry, and calculus. They are also used in real-world applications, such as physics, engineering, and economics.

Q: Can I use the properties of equality to solve systems of equations?

A: Yes, you can use the properties of equality to solve systems of equations. However, you may need to use additional techniques, such as substitution or elimination, to solve the system.

Q: Are there any other properties of equality that I should know about?

A: Yes, there are several other properties of equality that you should know about, including:

  • The division property of equality: If we divide both sides of an equation by the same non-zero value, the equation remains true.
  • The identity property of equality: If we add, subtract, or multiply both sides of an equation by the same value, the equation remains true.
  • The inverse property of equality: If we add, subtract, or multiply both sides of an equation by the additive or multiplicative inverse of a value, the equation remains true.

Conclusion

In conclusion, the properties of equality are fundamental concepts in mathematics that allow us to manipulate equations without changing their solutions. By understanding and applying these properties, you can solve linear equations and make sense of the world around you. Whether it's solving a simple equation or modeling complex real-world phenomena, the properties of equality are essential tools that every mathematician and scientist should know.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Linear Algebra and Its Applications" by Gilbert Strang
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

  • Khan Academy: Linear Equations
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram MathWorld: Linear Equations