Problem 16 - College Algebra

Question

solve and check each linear equation. $$ \begin{array}{l} 45-[4-2 y-4(y+7)]- \\ -4(1+3 y)-[4-3(y+2)-2(2 y-5)] \end{array} $$

Step-by-Step Solution

Answer

The solution for \(y\) is \(\frac{5}{3}\)

Step 1: Simplify inner brackets

First we start by simplifying the equation within the innermost brackets. \nSo, \( 45 - [4 - 2y - 4(y + 7)] - 4(1 + 3y) - [4 - 3(y + 2) - 2(2y - 5)] = \)\( 45 - [4 - 2y - 4y - 28] - 4 - 12y - [4 - 3y - 4 - 4y + 10] \)

Step 2: Combine like terms and simplify

Next, we combine like terms and simplify each section of the equation inside the brackets. \nSo, \( 45 - [- 6y - 32] - 4 - 12y - 3y + 6 = \)\( 45 - (-6y + 32) - 4 - 12y - 3y + 6 = \) \( 45 + 6y - 32 - 4 - 12y - 3y + 6 \)

Step 3: Combine like terms again

Now we combine like terms again and simplify the equation: \n\( 45 + 6y - 32 - 4 - 12y - 3y + 6 = 15 - 9y \)

Step 4: Solve for \(y\)

Finally, isolate y to one side of the equation to find its value. So, \n\( 15 - 9y = 0 \Rightarrow 9y = 15 \Rightarrow y = 15/9 = 5/3 \)

Key Concepts

Simplifying Expressions

Simplifying expressions is all about reducing a complex mathematical phrase into a simpler form. When faced with a linear equation like the one in our exercise, the first step is to tackle the inner brackets.
Look for operations inside the brackets and perform them first. This is important because it follows the order of operations known as PEMDAS/BODMAS.

• Identify terms inside the innermost brackets.
• Perform any addition, subtraction, multiplication, or division as needed.
• Simplify the expression within each bracket until no further operations are possible inside them.

In our exercise, the terms within the brackets were simplified first by distributing the multiplication, taking care of signs and subtraction operations.

Combining Like Terms

Combining like terms means bringing together terms in an expression that have the same variable raised to the same power.
This helps in streamlining the expression into fewer terms which are easier to work with. In the context of our exercise:

• Look for the terms with the same variable like 'y' in our equation.
• Add or subtract these terms to combine them.
• Edit the expression to reflect these changes, further simplifying it.

By combining like terms in our exercise, we were able to bring together values of 'y' to effectively manage the equation. This is crucial before you move on to solving the equation to find the unknown variable.

Isolation of Variables

Isolation of variables is the process of manipulating an equation so that the variable you're solving for is on one side alone. For linear equations, the goal is to get the variable by itself to find its value. Here's how we did it in our problem:

• Rearrange the equation to bring all terms involving the variable onto one side.
• Perform arithmetic operations to isolate the variable, simplifying any coefficients if necessary.
• Once isolated, solve for the variable by performing the required division or multiplication.

In our step-by-step solution, this meant rearranging the terms so that 'y' was on one side of the equation. Then, we divided by the coefficient to solve for 'y', ultimately finding that it equals \( \frac{5}{3} \).

Step-by-Step Solution

A step-by-step solution can turn a daunting problem into a manageable task. Our exercise was broken down methodically, allowing the solution to unfold clearly. A clear method shows how each operation affects the next:

• First, simplify the expression by dealing with the innermost brackets.
• Next, combine like terms, making the equation cleaner and focusing on the variable.
• Then, isolate the variable, arranging the equation to solve it effectively.
• Finally, perform the arithmetic to find the unknown variable's value.

This structured approach ensures no step is skipped and makes it easier to identify and correct mistakes. By seeing each step as a small goal towards a larger solution, you can effectively navigate through complex linear equations.