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given that 3x−7=−6y3x-7=-6y,evaluate 3x−74(x+4y)−283\frac{3x-7}{4(x+4y)-\frac{28}{3}}

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Full solution

Q. given that 3x−7=−6y3x-7=-6y,evaluate 3x−74(x+4y)−283\frac{3x-7}{4(x+4y)-\frac{28}{3}}
  1. Simplify Denominator Expression: First, we need to simplify the expression in the denominator 4(x+4y)−2834(x+4y)-\frac{28}{3}. We can distribute the 44 into the terms inside the parentheses.\newlineCalculation: 4(x+4y)−283=4x+16y−2834(x+4y)-\frac{28}{3} = 4x + 16y - \frac{28}{3}
  2. Substitute Value in Numerator: Next, we need to substitute the value of 3x−73x-7 with −6y-6y in the numerator, as given by the equation 3x−7=−6y3x-7=-6y.\newlineCalculation: The numerator becomes −6y-6y.
  3. Substitute Value in Denominator: Now, we substitute the value of 3x−73x-7 in the denominator with −6y-6y as well, since the expression 4(x+4y)4(x+4y) contains the term xx which is related to yy by the given equation.\newlineCalculation: The denominator becomes 4(−6y/3+4y)−28/34(-6y/3 + 4y) - 28/3.
  4. Combine Like Terms: We simplify the denominator further by combining like terms and simplifying the fraction −6y/3-6y/3.\newlineCalculation: The denominator simplifies to −8y−28/3-8y - 28/3.
  5. Find Common Denominator: Now we have the expression −6y/(−8y−28/3)-6y / (-8y - 28/3). We can simplify this by finding a common denominator for the terms in the denominator.\newlineCalculation: The common denominator is 33, so we rewrite −8y-8y as −24y/3-24y/3 and combine it with −28/3-28/3 to get −24y/3−28/3-24y/3 - 28/3.
  6. Combine Denominator Terms: We combine the terms in the denominator to get a single fraction.\newlineCalculation: The denominator becomes (−24y−28)/3(-24y - 28)/3.
  7. Multiply by Reciprocal: Now we have the simplified expression −6y/((−24y−28)/3)-6y / ((-24y - 28)/3). To divide by a fraction, we multiply by its reciprocal.\newlineCalculation: The expression becomes −6y×(3/(−24y−28))-6y \times (3/(-24y - 28)).
  8. Cancel Out Factors: We can simplify the expression by canceling out a factor of −6-6 in the numerator and denominator.\newlineCalculation: The expression simplifies to −6y×(3−4y×6−28)-6y \times \left(\frac{3}{-4y \times 6 - 28}\right).
  9. Cancel Out −6-6: We can now cancel out the −6-6 in the numerator with the −6-6 in the denominator.\newlineCalculation: The expression simplifies to y×(3−4y−28)y \times \left(\frac{3}{-4y - 28}\right).
  10. Final Simplified Expression: Finally, we have the simplified expression y×(3−4y−28)y \times \left(\frac{3}{-4y - 28}\right). This is the evaluated form of the original expression given the condition 3x−7=−6y3x-7=-6y.

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