Problem 10 - Algebra A Combined Function

Question

Solve each equation. See Examples 1 and \(2 .\) $$ -4(n-4)-23=-7 $$

Step-by-Step Solution

Answer

The solution is \(n = 0\).

Step 1: Distribute -4

Start by applying the distributive property to the left side of the equation. Multiply \(-4\) by each term inside the parentheses. This turns the equation into:\[-4(n) + 4(4) - 23 = -7\] which simplifies to:\[-4n + 16 - 23 = -7\].

Step 2: Simplify the Equation

Combine the constant terms on the left side of the equation. Simplify \(16 - 23\) to get:\[-4n - 7 = -7\].

Step 3: Isolate the Variable Term

Add \(7\) to both sides of the equation to eliminate the constant term on the left side:\[-4n - 7 + 7 = -7 + 7\] which simplifies to:\[-4n = 0\].

Step 4: Solve for the Variable

Divide both sides by \(-4\) to solve for \(n\):\[n = \frac{0}{-4}\] which simplifies to:\[n = 0\].

Key Concepts

Distributive Property

When solving linear equations, the distributive property is a crucial tool. It allows you to remove parentheses by distributing a multiplier across the terms inside.
This property states that for any numbers or variables, a(b + c) = ab + ac.
In the problem you are working on, this property helps simplify the equation.
To apply it to -4(n-4), you distribute -4 to both n and -4 inside the parentheses.
It becomes -4(n) + (-4)(4).
This distribution helps to get rid of the parentheses and express the equation in a simpler form without losing any information.
It's a very basic yet powerful algebraic tool that makes handling equations much easier.

Isolating Variables

Once you've simplified the equation using distributive property, the next step is to isolate the variable.
To "isolate" means to get the variable, like n, alone on one side of the equation.
This typically involves several operations:

• Addition or subtraction to eliminate constants
• Multiplication or division to remove coefficients

In the provided equation, -4n - 7 = -7, you aim to make -4n alone on the left side.
By adding 7 to both sides, the equation simplifies to -4n = 0.
This step moves towards revealing the value of n, getting us closer to the solution.

Simplifying Equations

Simplifying is all about making the equation easier to handle.
Many algebra problems begin complex, with multiple terms and operations.
Simplification gets rid of extra elements and combines like terms, making solving much more straightforward.
In your exercise, this process is used right after applying the distributive property and before isolating the variable.

You combined constants 16 and -23 to get -7, reducing the equation to fewer terms.
This cleanup operation often involves basic arithmetic such as addition or subtraction and helps prevent errors during calculations.
It is important to regularly simplify to keep track of your operations and ensure clarity in solving any equation.