If $4 x-2 y=4$ and $x y=\frac{6}{4}$, then find the value of $x^3-\frac{y^3}{8}$
Answer (2)
The value of x 3 − 8 y 3 for the given equations is 4 13 . This was determined by substituting and solving for x and y based on the provided equations. Both found pairs of solutions yield the same result, confirming its accuracy.
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Simplify the first equation to express y in terms of x : y = 2 x − 2 .
Substitute this expression into the second equation to obtain a quadratic equation in x : 4 x 2 − 4 x − 3 = 0 .
Solve the quadratic equation to find two possible values for x : x = 2 3 and x = − 2 1 .
For each value of x , find the corresponding value of y and compute x 3 − 8 y 3 , which yields 4 13 in both cases. Therefore, the final answer is 4 13 .
Explanation
Problem Analysis We are given two equations: 4 x − 2 y = 4 and x y = 4 6 . Our goal is to find the value of the expression x 3 − 8 y 3 .
Simplifying the First Equation First, let's simplify the first equation: 4 x − 2 y = 4 . We can divide both sides by 2 to get 2 x − y = 2 . From this, we can express y in terms of x : y = 2 x − 2 .
Substitution Now, substitute y = 2 x − 2 into the second equation x y = 4 6 : x ( 2 x − 2 ) = 4 6 . This simplifies to 2 x 2 − 2 x = 2 3 .
Forming a Quadratic Equation To solve for x , we can multiply both sides of the equation 2 x 2 − 2 x = 2 3 by 2 to eliminate the fraction: 4 x 2 − 4 x = 3 . Rearranging the terms, we get a quadratic equation: 4 x 2 − 4 x − 3 = 0 .
Solving the Quadratic Equation We can solve the quadratic equation 4 x 2 − 4 x − 3 = 0 using the quadratic formula: x = 2 a − b ± b 2 − 4 a c , where a = 4 , b = − 4 , and c = − 3 . Plugging in these values, we get: x = 2 ( 4 ) 4 ± ( − 4 ) 2 − 4 ( 4 ) ( − 3 ) = 8 4 ± 16 + 48 = 8 4 ± 64 = 8 4 ± 8 . So, x = 8 4 + 8 = 8 12 = 2 3 or x = 8 4 − 8 = 8 − 4 = − 2 1 .
Finding the Values of y Now, we find the corresponding values of y for each value of x using the equation y = 2 x − 2 . If x = 2 3 , then y = 2 ( 2 3 ) − 2 = 3 − 2 = 1 . If x = − 2 1 , then y = 2 ( − 2 1 ) − 2 = − 1 − 2 = − 3 .
Calculating the Expression We have two pairs of solutions: ( x , y ) = ( 2 3 , 1 ) and ( x , y ) = ( − 2 1 , − 3 ) . Now, we calculate x 3 − 8 y 3 for each pair. If x = 2 3 and y = 1 , then x 3 − 8 y 3 = ( 2 3 ) 3 − 8 1 3 = 8 27 − 8 1 = 8 26 = 4 13 . If x = − 2 1 and y = − 3 , then x 3 − 8 y 3 = ( − 2 1 ) 3 − 8 ( − 3 ) 3 = − 8 1 − 8 − 27 = − 8 1 + 8 27 = 8 26 = 4 13 .
Final Answer In both cases, we get the same value for x 3 − 8 y 3 , which is 4 13 .
Examples
Imagine you're designing a rectangular garden where the length and width must satisfy certain conditions related to area and perimeter. The equations provided are similar to constraints you might encounter in such a design problem. By solving these equations, you determine the dimensions of the garden. Then, you might want to calculate a related quantity, such as the volume of soil needed for a specific depth, which could be represented by the expression we evaluated, x 3 − 8 y 3 . This exercise demonstrates how algebraic solutions can be applied to practical design and resource calculation problems.
