Problem 88 - Algebra and Trigonometry

Question

Evaluate \(x^{2}-(x y-y)\) for \(x\) satisfying \(\frac{13 x-6}{4}=5 x+2\) and \(y\) satisfying \(5-y=7(y+4)+1\)

Step-by-Step Solution

Answer

The value of the expression \(x^{2}-(x y-y)\) for the given \(x\) and \(y\) is \(7\).

Step 1: Solve for \(x\)

Solve the equation \(\frac{13 x-6}{4}=5 x+2\) for \(x\).\nFirst, multiply every term by \(4\) to clear the fraction, then bring similar terms together and solve:\n\(13x - 6 = 20x + 8\)\n\(8+6 = 20x-13x\)\n\(\implies x = 2\)

Step 2: Solve for \(y\)

Solve the equation \(5-y=7(y+4)+1\) for \(y\).\nFirst, distribute the \(7\) on the right side, then bring similar terms together and solve: \n\(5 - y = 7y + 28 + 1\)\n\(5 - 28 - 1 = 7y + y\)\n\(\implies y = -3\)

Step 3: Evaluate the expression

Substitute \(x = 2\) and \(y = -3\) into the expression \(x^{2}-(x y-y)\) and simplify to find the result:\n\(= 2^{2} - (2* -3 - (-3)) \) \n\(= 4 - (-6 +3) \) \n\(= 4 - (-3) \) \n\(= 4 + 3 = 7\)