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f) {[(x+y)/(2)-(x-y)/(3)=3],[(x+2y)/(3)-(x-2y)/(4)=3]:}

f) {x+y2−x−y3=3x+2y3−x−2y4=3 \left\{\begin{array}{l}\frac{x+y}{2}-\frac{x-y}{3}=3 \\ \frac{x+2 y}{3}-\frac{x-2 y}{4}=3\end{array}\right.

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Q. f) {x+y2−x−y3=3x+2y3−x−2y4=3 \left\{\begin{array}{l}\frac{x+y}{2}-\frac{x-y}{3}=3 \\ \frac{x+2 y}{3}-\frac{x-2 y}{4}=3\end{array}\right.
  1. Write Equations: First, let's write down the system of equations:\newlinex+y2−x−y3=3 \frac{x+y}{2}-\frac{x-y}{3}=3 , x+2y3−x−2y4=3 \frac{x+2y}{3}-\frac{x-2y}{4}=3
  2. Solve First Equation: Now, let's solve the first equation for xx and yy:x+y2−x−y3=3\frac{x+y}{2}-\frac{x-y}{3}=3To clear the fractions, multiply everything by the least common multiple of 22 and 33, which is 66:6(x+y2)−6(x−y3)=6×36\left(\frac{x+y}{2}\right) - 6\left(\frac{x-y}{3}\right) = 6\times 3
  3. Simplify First Equation: Simplify the equation: 3(x+y)−2(x−y)=183(x+y) - 2(x-y) = 18
  4. Combine Terms: Distribute the 33 and the 22:3x+3y−2x+2y=183x + 3y - 2x + 2y = 18
  5. Solve Second Equation: Combine like terms: x+5y=18x + 5y = 18
  6. Simplify Second Equation: Now, let's solve the second equation for xx and yy:x+2y3−x−2y4=3\frac{x+2y}{3}-\frac{x-2y}{4}=3To clear the fractions, multiply everything by the least common multiple of 33 and 44, which is 1212:12(x+2y3)−12(x−2y4)=12×312\left(\frac{x+2y}{3}\right) - 12\left(\frac{x-2y}{4}\right) = 12\times 3
  7. Combine Terms: Simplify the equation: 4(x+2y)−3(x−2y)=364(x+2y) - 3(x-2y) = 36
  8. Combine Terms: Simplify the equation:\newline4(x+2y)−3(x−2y)=364(x+2y) - 3(x-2y) = 36Distribute the 44 and the 33:\newline4x+8y−3x+6y=364x + 8y - 3x + 6y = 36
  9. Combine Terms: Simplify the equation:\newline4(x+2y)−3(x−2y)=364(x+2y) - 3(x-2y) = 36Distribute the 44 and the 33:\newline4x+8y−3x+6y=364x + 8y - 3x + 6y = 36Combine like terms:\newlinex+14y=36x + 14y = 36\newlineOops, made a mistake while combining the terms. It should be:\newlinex+14y=36x + 14y = 36 should be x+14y=36x + 14y = 36, but I accidentally added the y's wrong. It should be 4x+8y−3x−6y=364x + 8y - 3x - 6y = 36, which simplifies to x+2y=36x + 2y = 36.

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