If 2x - 3y - 7 = 0, then what is the value of 8x 3 - 36x 2y + 54xy 2 - 27y 3 - 340 ?
3
Understanding the Problem
The question asks us to find the value of a given algebraic expression: \(8x^3 - 36x^2y + 54xy^2 - 27y^3 - 340\). We are also provided with a linear equation relating \(x\) and \(y\): \(2x - 3y - 7 = 0\).
Our strategy will be to simplify the given equation to find a relationship between \(x\) and \(y\), and then look for ways to simplify the expression whose value we need to find. The structure of the expression \(8x^3 - 36x^2y + 54xy^2 - 27y^3\) looks similar to a known algebraic identity.
Analyzing the Given Equation
The given equation is:
\(2x - 3y - 7 = 0\)
We can rearrange this equation to isolate the term \(2x - 3y\):
\(2x - 3y = 7\)
This gives us a direct value for the expression \((2x - 3y)\), which we can use later.
Identifying the Algebraic Identity
Let's look at the first part of the expression whose value we need to find: \(8x^3 - 36x^2y + 54xy^2 - 27y^3\). This polynomial involves cubic terms of \(x\) and \(y\) and mixed terms with coefficients that look familiar.
Recall the binomial expansion for a cube: \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\).
Let's compare this with the given expression. Suppose \(a = 2x\) and \(b = 3y\).
- \(a^3 = (2x)^3 = 8x^3\)
- \(3a^2b = 3(2x)^2(3y) = 3(4x^2)(3y) = 36x^2y\)
- \(3ab^2 = 3(2x)(3y)^2 = 3(2x)(9y^2) = 54xy^2\)
- \(b^3 = (3y)^3 = 27y^3\)
Substituting these into the identity \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\):
\((2x - 3y)^3 = (2x)^3 - 3(2x)^2(3y) + 3(2x)(3y)^2 - (3y)^3\)
\( = 8x^3 - 36x^2y + 54xy^2 - 27y^3\)
This matches the first part of the expression given in the question.
Evaluating the Expression
The expression we need to evaluate is \(8x^3 - 36x^2y + 54xy^2 - 27y^3 - 340\).
Using the identity we just confirmed, we can rewrite this as:
\((2x - 3y)^3 - 340\)
From the given equation, we found that \(2x - 3y = 7\).
Substitute the value of \((2x - 3y)\) into the expression:
\((7)^3 - 340\)
Now, calculate the value:
\(7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343\)
So, the expression becomes:
\(343 - 340 = 3\)
The value of the expression \(8x^3 - 36x^2y + 54xy^2 - 27y^3 - 340\) is 3.
Comparing with Options
Let's check the given options:
- Option 1: -1
- Option 2: 0
- Option 3: 1
- Option 4: 3
Our calculated value is 3, which matches Option 4.
Revision Table: Relevant Algebraic Identities
| Identity | Expansion |
|---|---|
| \((a+b)^3\) | \(a^3 + 3a^2b + 3ab^2 + b^3\) |
| \((a-b)^3\) | \(a^3 - 3a^2b + 3ab^2 - b^3\) |
| \(a^3 + b^3\) | \((a+b)(a^2 - ab + b^2)\) |
| \(a^3 - b^3\) | \((a-b)(a^2 + ab + b^2)\) |
Additional Information: Manipulating Equations
The first step in solving this problem was rearranging the linear equation \(2x - 3y - 7 = 0\) to get \(2x - 3y = 7\). This is a fundamental skill in algebra called isolating a term or variable.
To isolate a term, you perform the same operation on both sides of the equality sign to maintain the balance. In this case, we added 7 to both sides of the equation \(2x - 3y - 7 = 0\):
\(2x - 3y - 7 + 7 = 0 + 7\)
Which simplifies to:
\(2x - 3y = 7\)
This allows us to substitute the value of the expression \((2x - 3y)\) directly into the more complex cubic expression, significantly simplifying the calculation.
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