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Chapter 0: Problem 25

Solve for the indicated variable. $$20-4[c-3-6(2 c+3)]=5(3 c-2)-[2(7 c-8)-4 c+7]$$

Short Answer

Expert verified
The value of \(c\) is \(-\frac{35}{13}\).

Step by step solution

01

Expand the Expressions

First, expand the expressions inside the brackets. In the expression \(-4[c-3-6(2c+3)]\), distribute the \(-6\) first to get \(-4[c - 3 - 12c - 18]\). Similarly, in the expression \(-[2(7c-8)-4c+7]\), distribute to get \(-[14c - 16 - 4c + 7]\).
02

Simplify Inside Brackets

After expanding, simplify each part: \(c - 3 - 12c - 18 = -11c - 21\) and \(14c - 16 - 4c + 7 = 10c - 9\). The expression becomes \(-4(-11c - 21)\) and \(-[10c - 9]\).
03

Distribute and Expand Again

Distribute the remaining constants: \(-4(-11c - 21) = 44c + 84\) and \(-[10c - 9] = -10c + 9\). Apply these to simplify the main expression: \(20 + 44c + 84 = 15c - 10 - 10c + 9\).
04

Combine Like Terms

Combine the constants and like terms on both sides: Left side: \(20 + 84 + 44c = 104 + 44c\)Right side: \(15c - 10c - 10 + 9 = 5c - 1\).
05

Set Equation and Solve for c

Set the simplified equation: \(104 + 44c = 5c - 1\)Subtract \(5c\) from both sides: \(104 + 39c = -1\)Subtract \(104\) from both sides: \(39c = -105\)Divide by \(39\) to isolate \(c\): \(c = \frac{-105}{39}\).
06

Simplify the Fraction

Simplify the fraction \(\frac{-105}{39}\). The greatest common divisor of 105 and 39 is 3. Divide both the numerator and the denominator by 3:\(c = \frac{-105 \div 3}{39 \div 3} = \frac{-35}{13}\).

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Key Concepts

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

Expanding Expressions
Expanding expressions is a fundamental process in algebra used to simplify problems and make solving them more straightforward. Imagine you have a group of terms inside parentheses with a number outside, like
  • \(-4[c-3-6(2c+3)]\)
To expand, multiply each term inside the parentheses by the number outside. This process is also known as distributing. For example, in
  • \(-4[c-3-6(2c+3)]\)
  • first, focus on \(-6(2c+3)\), handing out the -6 to each term inside, making it \(-12c - 18\). Then distribute \(-4\) across everything inside the new brackets.
Expanding expressions is like unpacking a box, making sure every item (or term) is accounted for. Expanding helps you see all parts of the equation clearly.
Simplifying Algebraic Expressions
Once you have expanded your expressions, it's important to tidy up the equation by simplifying it. Simplification involves merging several terms into fewer ones, essentially making the expression less cluttered and easier to understand.After expanding, you'll notice that some parts of the expression can be combined. For instance, in
  • \(c - 3 - 12c - 18\)
you can simplify by gathering like terms鈥攖hose terms that share the same variable or are constants. Simplifying this gives you
  • \(-11c - 21\).
Do the same for terms on the opposite side, combining constants with constants and variable terms with variable terms. This will help you form a more manageable equation to solve.
Combining Like Terms
Combining like terms is an essential part of simplifying algebraic expressions. When expressions are expanded, they often include both like and unlike terms. Like terms have the same variable raised to the same power.For example, in
  • \(44c + 84 = 15c - 10 - 10c + 9\),
you have terms like
  • \(44c\) and \(15c - 10c\); these are like terms because they share the same variable \(c\).
When combining these, just add or subtract the coefficients (the numbers in front of the variable) while retaining the common variable, for instance,
  • \(44c\) on one side becomes \(44c\),
  • and \(15c - 10c\) simplifies to \(5c\).
With combining, focus on reducing expressions to a simpler form, making it easier to isolate and solve for the variable at hand.

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Most popular questions from this chapter

Find the number \(a\) for which \(y=2\) is a solution of the equation \(y-a=y+5-3 a y\)

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