Use Properties To Rewrite The Given Equation. Which Equations Have The Same Solution As The Equation$\frac{3}{5} X + \frac{2}{3} + X = \frac{1}{2} - \frac{1}{5} X$Select Three Options:A. $\frac{8}{5} X + \frac{2}{3} = \frac{1}{2} -

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Introduction

In mathematics, equations are a fundamental concept that helps us solve problems and understand various mathematical relationships. When dealing with equations, it's essential to be able to rewrite them in different forms to simplify the solution process. In this article, we will explore how to use properties to rewrite the given equation and identify which equations have the same solution as the original equation.

The Original Equation

The given equation is:

35x+23+x=12βˆ’15x\frac{3}{5} x + \frac{2}{3} + x = \frac{1}{2} - \frac{1}{5} x

Our goal is to rewrite this equation using properties and identify which equations have the same solution as the original equation.

Step 1: Combine Like Terms

The first step in rewriting the equation is to combine like terms. We can start by combining the terms with the variable x:

35x+x=85x\frac{3}{5} x + x = \frac{8}{5} x

Now, the equation becomes:

85x+23=12βˆ’15x\frac{8}{5} x + \frac{2}{3} = \frac{1}{2} - \frac{1}{5} x

Step 2: Move All Terms to One Side

Next, we need to move all terms to one side of the equation. We can do this by subtracting 15x\frac{1}{5} x from both sides:

85x+15x=12βˆ’23\frac{8}{5} x + \frac{1}{5} x = \frac{1}{2} - \frac{2}{3}

Simplifying the left-hand side, we get:

95x=12βˆ’23\frac{9}{5} x = \frac{1}{2} - \frac{2}{3}

Step 3: Simplify the Right-Hand Side

Now, we need to simplify the right-hand side of the equation. We can do this by finding a common denominator for the fractions:

12βˆ’23=36βˆ’46\frac{1}{2} - \frac{2}{3} = \frac{3}{6} - \frac{4}{6}

Simplifying further, we get:

βˆ’16-\frac{1}{6}

So, the equation becomes:

95x=βˆ’16\frac{9}{5} x = -\frac{1}{6}

Step 4: Solve for x

Finally, we can solve for x by multiplying both sides of the equation by the reciprocal of 95\frac{9}{5}:

x=βˆ’16Γ—59x = -\frac{1}{6} \times \frac{5}{9}

Simplifying, we get:

x=βˆ’554x = -\frac{5}{54}

Which Equations Have the Same Solution?

Now that we have rewritten the original equation, we need to identify which equations have the same solution. Let's consider the three options:

A. 85x+23=12βˆ’15x\frac{8}{5} x + \frac{2}{3} = \frac{1}{2} - \frac{1}{5} x

B. 95x=12βˆ’23\frac{9}{5} x = \frac{1}{2} - \frac{2}{3}

C. x=βˆ’554x = -\frac{5}{54}

We can see that option B is equivalent to the rewritten equation we obtained in Step 3. Therefore, option B has the same solution as the original equation.

Conclusion

In this article, we learned how to use properties to rewrite the given equation and identify which equations have the same solution. We started by combining like terms, moved all terms to one side, simplified the right-hand side, and finally solved for x. We also identified which equations have the same solution as the original equation. By following these steps, you can rewrite equations using properties and solve problems with ease.

Additional Tips and Tricks

  • When rewriting equations, it's essential to be careful with the order of operations.
  • Make sure to simplify the right-hand side of the equation by finding a common denominator.
  • When solving for x, multiply both sides of the equation by the reciprocal of the coefficient of x.

Q: What are the properties used to rewrite equations?

A: The properties used to rewrite equations include the commutative property, associative property, distributive property, and the properties of exponents. These properties help us simplify and rearrange equations to make them easier to solve.

Q: How do I combine like terms when rewriting an equation?

A: To combine like terms, we need to identify the terms with the same variable and coefficient. We can then add or subtract these terms to simplify the equation. For example, in the equation 35x+x=85x\frac{3}{5} x + x = \frac{8}{5} x, we can combine the terms with the variable x by adding them together.

Q: What is the difference between the commutative and associative properties?

A: The commutative property states that the order of the terms in an equation does not change the result. For example, in the equation a+b=b+aa + b = b + a, the order of the terms a and b does not change the result. The associative property states that the order in which we perform operations in an equation does not change the result. For example, in the equation (a+b)+c=a+(b+c)(a + b) + c = a + (b + c), the order in which we perform the addition does not change the result.

Q: How do I simplify the right-hand side of an equation?

A: To simplify the right-hand side of an equation, we need to find a common denominator for the fractions. We can then add or subtract the fractions to simplify the equation. For example, in the equation 12βˆ’23\frac{1}{2} - \frac{2}{3}, we can find a common denominator of 6 and simplify the equation to βˆ’16-\frac{1}{6}.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and denominator. For example, the reciprocal of 35\frac{3}{5} is 53\frac{5}{3}.

Q: How do I solve for x in an equation?

A: To solve for x in an equation, we need to isolate the variable x on one side of the equation. We can do this by adding or subtracting terms from both sides of the equation, and then multiplying or dividing both sides by the coefficient of x. For example, in the equation 95x=βˆ’16\frac{9}{5} x = -\frac{1}{6}, we can solve for x by multiplying both sides by the reciprocal of 95\frac{9}{5}.

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

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

  • Not combining like terms
  • Not simplifying the right-hand side of the equation
  • Not isolating the variable x on one side of the equation
  • Not using the correct properties to simplify the equation

Q: How can I practice rewriting equations using properties?

A: You can practice rewriting equations using properties by working through examples and exercises in a math textbook or online resource. You can also try rewriting equations on your own and then checking your work to make sure you are using the correct properties.

Q: What are some real-world applications of rewriting equations using properties?

A: Rewriting equations using properties has many real-world applications, including:

  • Solving problems in physics and engineering
  • Modeling population growth and decay
  • Analyzing financial data and making investment decisions
  • Solving problems in computer science and programming

By mastering the skills of rewriting equations using properties, you can solve a wide range of problems and make informed decisions in many areas of life.