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Chapter 1: Problem 88

Solve the equation for the indicated variable. \(a-2[b-3(c-x)]=6 ; \quad\) for \(x\)

Short Answer

Expert verified
\(x = \frac{a - 2b + 6c - 6}{6}\)

Step by step solution

01

Distribute the Minus Sign

Start by distributing the minus sign across the terms inside the brackets. This means multiplying each term inside the brackets by -1. The expression becomes:\[a - 2b + 6(c-x) = 6\]
02

Distribute the 6

Now, distribute the 6 in front of the parenthesis across the terms inside it. This means multiplying both terms inside by 6. The expression transforms into:\[a - 2b + 6c - 6x = 6\]
03

Move Terms Involving x to One Side

Reorder the equation so that all terms involving \(x\) are on one side and constants on the other. Add \(6x\) to both sides and subtract \(a\), \(-2b\), and \(6c\) from both sides:\[6x = a - 2b + 6c - 6\]
04

Isolate x

Solve for \(x\) by dividing each term in the equation by 6:\[x = \frac{a - 2b + 6c - 6}{6}\]

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Key Concepts

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

Equation Solving
Solving an equation involves finding the value of the variable that makes the equation true. Think of it as a balanced scale, where both sides need to weigh the same. When you solve equations, you're trying to keep the balance while performing operations.
  • Start with simplifying the equation if necessary. This can involve removing brackets or combining like terms.
  • Next, systematically work to isolate the variable by ensuring that it is left on one side of the equation alone.
Remember, whatever you do to one side of the equation, you must do to the other to maintain balance. Always check your solution by plugging it back into the original equation to see if both sides are equal.
Variable Isolation
Variable isolation is the process of manipulating an equation so the variable in question stands alone on one side. In essence, this is like peeling away layers to uncover what you're solving for鈥攊n this case, the variable x. To isolate a variable:
  • Move all other terms to the opposite side of the equation using addition or subtraction.
  • Undo any operations applied to the variable, like multiplication or division.
In the example exercise, our goal was to have all expressions involving x on one side. We achieved this by moving terms around and simplifying the equation systematically. Once isolated, determining the variable's value becomes much more straightforward.
Algebraic Manipulation
Algebraic manipulation is the set of techniques used to rearrange and simplify equations and expressions. This involves using arithmetic operations to transform equations while maintaining their equality. Key strategies include:
  • Distributing multiplication over addition or subtraction to eliminate parentheses.
  • Combining like terms to simplify expressions.
  • Reversing operations by applying the opposite operation to eliminate unnecessary terms.
In the given solution, algebraic manipulation involved distributing terms, rearranging the equation by moving terms, and finally dividing by a coefficient to solve for the variable. These techniques not only simplify the problem but also guide us to find the variable's value efficiently and accurately.

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Most popular questions from this chapter

Recall that the symbol \(\overline{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$ \overline{z w}=\overline{z} \cdot \overline{w} $$

For the complex number \(3+4 i\) the real part is _________ the imaginary part is ____________

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \frac{1-\sqrt{-1}}{1+\sqrt{-1}} $$

Evaluate the expression and write the result in the form a bi. $$ (2 i)^{4} $$

Find the real and imaginary parts of the complex number. $$ \sqrt{3}+\sqrt{-4} $$
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