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Solve the equations and check your answer in the each case:
`(x-2)/(3)+(x-3)/(4)=(x-1)/(2)`

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The correct Answer is:
To solve the equation \(\frac{x-2}{3} + \frac{x-3}{4} = \frac{x-1}{2}\), we will follow these steps: ### Step 1: Find a common denominator The denominators in the equation are 3, 4, and 2. The least common multiple (LCM) of these numbers is 12. ### Step 2: Rewrite each term with the common denominator We can rewrite the equation as follows: \[ \frac{4(x-2)}{12} + \frac{3(x-3)}{12} = \frac{6(x-1)}{12} \] ### Step 3: Combine the left-hand side Now that all terms have the same denominator, we can combine the left-hand side: \[ \frac{4(x-2) + 3(x-3)}{12} = \frac{6(x-1)}{12} \] ### Step 4: Remove the denominator Since both sides have the same denominator (12), we can multiply through by 12 to eliminate the denominator: \[ 4(x-2) + 3(x-3) = 6(x-1) \] ### Step 5: Expand each term Now, we will expand each term: \[ 4x - 8 + 3x - 9 = 6x - 6 \] ### Step 6: Combine like terms Combine the \(x\) terms and the constant terms: \[ (4x + 3x) - 17 = 6x - 6 \] This simplifies to: \[ 7x - 17 = 6x - 6 \] ### Step 7: Isolate \(x\) Now, we will isolate \(x\) by moving \(6x\) to the left side and \(17\) to the right side: \[ 7x - 6x = 17 - 6 \] This simplifies to: \[ x = 11 \] ### Step 8: Check the solution To verify our solution, we substitute \(x = 11\) back into the original equation: - Left-hand side (LHS): \[ \frac{11-2}{3} + \frac{11-3}{4} = \frac{9}{3} + \frac{8}{4} = 3 + 2 = 5 \] - Right-hand side (RHS): \[ \frac{11-1}{2} = \frac{10}{2} = 5 \] Since LHS = RHS = 5, our solution is verified. ### Final Answer The solution to the equation is \(x = 11\).
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Knowledge Check

  • Solve the equations and check your answer in the case: 5(x+4)=35

    A
    z = 2
    B
    z = 4
    C
    z = 3
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    z = 5
  • Solve the equations and check your answer in the case: 7-2(5-3x)=4(x-3)+5

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    x = -3
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    x = -2
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    x = -4
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    x = -5
  • Solve the equations and check your answer in the case: (2x+3)/(3+x)=(3)/(2)

    A
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    x = 1
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