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Chapter 1: Problem 59

Solve each equation for \(x\) in terms of the other letters. \(a^{2}(a-x)=b^{2}(b+x)-2 a b x,\) where \(a \neq b\)

Short Answer

Expert verified
\(x = \frac{b^3 - a^3}{2ab - a^2 - b^2}\)

Step by step solution

01

Expand Both Sides

First, expand the left-hand side and right-hand side of the equation. For the left-hand side, distribute the term \(a^2\): \[a^2(a - x) = a^3 - a^2 x.\]For the right-hand side, expand each section: \[b^2(b + x) = b^3 + b^2 x.\]Thus, the equation becomes: \[a^3 - a^2 x = b^3 + b^2 x - 2abx.\]
02

Combine Like Terms

Move all terms involving \(x\) to one side and constant terms to the other. Gather the \(x\) terms on the left and constants on the right:\[-a^2 x - b^2 x + 2abx = b^3 - a^3.\]
03

Factor Out x

Factor \(x\) out of the terms on the left side:\[x(-a^2 - b^2 + 2ab) = b^3 - a^3.\]
04

Solve for x

Isolate \(x\) by dividing both sides by the coefficient of \(x\):\[x = \frac{b^3 - a^3}{-a^2 - b^2 + 2ab}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expanding Expressions
Expanding expressions involves removing parentheses by distributing terms across the expression. This step helps simplify the equation to further solve for the variable. It's crucial for breaking down complex equations into simpler, manageable parts. Consider the expression on the left-hand side of the given equation: \(a^2(a - x)\) To expand, distribute \(a^2\) to both \(a\) and \(-x\). The expanded form becomes: \[a^2(a - x) = a^3 - a^2 x\]. Similarly, for the right side, expand \(b^2(b + x)\) as follows: \[b^2(b + x) = b^3 + b^2 x\]. Expanding clarifies the terms involved and sets the stage for combining and simplifying.
Factoring
Factoring is the process of breaking down expressions into simpler factors that multiply to form the original expression. It's a powerful tool for solving equations, particularly when you're dealing with polynomials. In our example, after rearranging terms, the equation becomes: \[-a^2 x - b^2 x + 2abx = b^3 - a^3\].Notice each term on the left contains \(x\). We can factor \(x\) out to simplify the equation to: \[x(-a^2 - b^2 + 2ab) = b^3 - a^3\]. By factoring out \(x\), the equation becomes easier to work with and helps us isolate \(x\) in the later steps.
Combining Like Terms
Combining like terms simplifies an equation by adding or subtracting terms with the same variables. This refinement balances and organizes the equation, making it easier to solve.After expanding both sides of our equation, ensure all \(x\) terms are on one side and constant terms on the other. We start with: \[a^3 - a^2 x = b^3 + b^2 x - 2abx\].Move terms to consolidate like terms: \[-a^2 x - b^2 x + 2abx = b^3 - a^3\]. It's a helpful step that supports factoring and further simplification of the equation.
Isolation of Variables
Isolation of variables is the key to solving an equation for a specific variable. After combining like terms and factoring, the next step is to isolate the variable you are interested in.Following the simplification of our initial equation, we have:\[x(-a^2 - b^2 + 2ab) = b^3 - a^3\]. To isolate \(x\), divide both sides by the expression \(-a^2 - b^2 + 2ab\):\[x = \frac{b^3 - a^3}{-a^2 - b^2 + 2ab}\].This division removes the coefficient of \(x\) from the left side, solving the equation for \(x\) in terms of other variables. Mastering this step provides the final value needed for understanding and solving similar equations.

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Most popular questions from this chapter

Explain why there are no real numbers that satisfy the equation \(\left|x^{2}+4 x\right|=-12\).

(a) Use a graphing utility to graph the equation. (b) Use a graphing utility, as in Example \(5,\) to estimate to one decimal place the \(x\) -intercepts. (c) Use algebra to determine the exact values for the \(x\) -intercepts. Then use a calculator to check that the answers are consistent with the estimates obtained in part (b). $$y=2 x^{2}+x-5$$

Determine the center and the radius for the circle. Also, find the \(y\) -coordinates of the points (if any) where the circle intersects the \(y\) -axis. $$3 x^{2}+3 y^{2}+5 x-4 y=1$$

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-4|<4$$

Imagine that you own a grove of orange trees, and suppose that from past experience you know that when 100 trees are planted, each tree will yield approximately 240 oranges per year. Furthermore, you've noticed that when additional trees are planted in the grove, the yield per tree decreases. Specifically, you have noted that the yield per tree decreases by about 20 oranges for each additional tree planted. (a) Let \(y\) denote the yield per tree when \(x\) trees are planted. Find a linear equation relating \(x\) and \(y\) Hint: You are given that the point (100,240) is on the line. What is given about \(\Delta y / \Delta x ?\) (b) Use the equation in part (a) to determine how many trees should be planted to obtain a yield of 400 oranges per tree. (c) If the grove contains 95 trees, what yield can you expect from each tree?
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