Solve And Check The Given Equation.${ \frac{x-2}{2} - 1 = \frac{x-7}{3} }$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. The Solution Set Is { □ \square □ }. (Type An Integer Or

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Introduction

In this article, we will be solving and checking a given equation. The equation is a linear equation involving fractions, and we will use algebraic techniques to solve it. We will also check our solution to ensure that it satisfies the original equation.

The Given Equation

The given equation is:

x−22−1=x−73\frac{x-2}{2} - 1 = \frac{x-7}{3}

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 6.

6(x−22−1)=6(x−73)6 \left( \frac{x-2}{2} - 1 \right) = 6 \left( \frac{x-7}{3} \right)

Step 2: Distribute the 6 to Both Terms Inside the Parentheses

Distributing the 6 to both terms inside the parentheses, we get:

3(x−2)−6=2(x−7)3(x-2) - 6 = 2(x-7)

Step 3: Expand and Simplify the Equation

Expanding and simplifying the equation, we get:

3x−6−6=2x−143x - 6 - 6 = 2x - 14

Step 4: Combine Like Terms

Combining like terms, we get:

3x−12=2x−143x - 12 = 2x - 14

Step 5: Add 12 to Both Sides

Adding 12 to both sides, we get:

3x=2x−23x = 2x - 2

Step 6: Subtract 2x from Both Sides

Subtracting 2x from both sides, we get:

x=−2x = -2

Step 7: Check the Solution

To check the solution, we need to plug x = -2 back into the original equation and see if it satisfies the equation.

−2−22−1=−2−73\frac{-2-2}{2} - 1 = \frac{-2-7}{3}

−42−1=−93\frac{-4}{2} - 1 = \frac{-9}{3}

−2−1=−3-2 - 1 = -3

−3=−3-3 = -3

Since the left-hand side and right-hand side are equal, we can conclude that x = -2 is indeed a solution to the equation.

Conclusion

In conclusion, we have solved and checked the given equation. The solution to the equation is x = -2, and we have verified that it satisfies the original equation.

Final Answer

The final answer is −2\boxed{-2}.

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Introduction

In our previous article, we solved and checked the given equation x−22−1=x−73\frac{x-2}{2} - 1 = \frac{x-7}{3}. In this article, we will answer some frequently asked questions related to solving and checking the equation.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: Why do we need to multiply both sides by the LCM?

A: We need to multiply both sides by the LCM to eliminate the fractions and make it easier to solve the equation.

Q: What is the LCM of 2 and 3?

A: The LCM of 2 and 3 is 6.

Q: How do we distribute the 6 to both terms inside the parentheses?

A: We distribute the 6 to both terms inside the parentheses by multiplying each term by 6.

Q: What is the next step after distributing the 6?

A: The next step is to expand and simplify the equation.

Q: How do we check the solution?

A: We check the solution by plugging the value of x back into the original equation and seeing if it satisfies the equation.

Q: What is the final answer to the equation?

A: The final answer to the equation is x = -2.

Q: Why is it important to check the solution?

A: It is important to check the solution to ensure that it satisfies the original equation and to verify that the solution is correct.

Q: What if the solution does not satisfy the original equation?

A: If the solution does not satisfy the original equation, then it is not a valid solution and we need to re-examine our work to find the correct solution.

Q: Can we use other methods to solve the equation?

A: Yes, we can use other methods to solve the equation, such as using algebraic manipulations or using a calculator to solve the equation.

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

A: Some common mistakes to avoid when solving the equation include:

  • Not eliminating the fractions
  • Not distributing the 6 correctly
  • Not checking the solution
  • Not re-examining the work if the solution does not satisfy the original equation

Conclusion

In conclusion, solving and checking the given equation requires careful attention to detail and a thorough understanding of algebraic manipulations. By following the steps outlined in this article, we can solve and check the equation with confidence.

Final Answer

The final answer is −2\boxed{-2}.