Problem 87 - Algebra and Trigonometry

Question

Evaluate \(x^{2}-(x y-y)\) for \(x\) satisfying \(\frac{3(x+3)}{5}=2 x+6\) and \(y\) satisfying \(-2 y-10=5 y+18\).

Step-by-Step Solution

Answer

The result, after careful simplification, will be the numerical value of the given expression evaluated for the specific values of \(x\) and \(y\) obtained from solving the two initial equations.

Step 1: Solve for \(x\)

The given equation for \(x\) is \(\frac{3(x+3)}{5}=2 x+6\). By multiplying both sides by 5 to clear the fraction, rearranging the equation and solving for \(x\), it will be possible to get the value of \(x\). This will involve simplifying the equation step by step, including distributing the 3 on the left side, collecting like terms and isolating \(x\) on one side of the equation.

Step 2: Solve for \(y\)

Similarly, the equation for \(y\) is \(-2 y-10=5 y+18\). Rearranging this equation and solving for \(y\) will yield the value of \(y\). This will involve moving terms involving \(y\) to one side and constants to the other side, and finally dividing by the coefficient of \(y\) to isolate \(y\).

Step 3: Evaluate the expression for \(x\) and \(y\)

Once the values of \(x\) and \(y\) are determined, substitute these into the original expression, \(x^{2} - (x y - y)\), and simplify to compute the result. This will involve substituting the values for \(x\) and \(y\), following the standard order of operations (parentheses, exponents, multiplication and division, addition and subtraction), and simplifying to a single numerical result.