Problem 33 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(-2(4-m)=10\)

Step-by-Step Solution

Answer

The solution for the equation is \(m = 9\).

Step 1: Distributing '-2' to both terms in the parentheses

We start off by applying the distributive property of multiplication over subtraction, which states that -2 times each term inside the parentheses should be calculated. This gives: \(-2*4 - (-2*m) = 10\), which simplifies to \(-8 + 2m = 10\).

Step 2: Rearranging terms

Now transfer '-8' from left side of equation to the right side by adding '8' on both sides: \(2m = 10 + 8\), which simplifies to \(2m = 18\).

Step 3: Solving for 'm'

Finally, isolate 'm' by dividing the entire equation by '2': \(m = 18 / 2\), which simplifies to \(m = 9\).

Key Concepts

Distributive Property

One of the foundational principles for solving linear equations is the distributive property. This property allows you to multiply each term within a set of parentheses by a factor outside the parentheses. It's like distributing the factor to both terms inside. For example, in the equation -2(4-m), we distribute -2 by multiplying it with each term inside the parentheses separately:

• First, multiply -2 by 4, resulting in -8.
• Then, multiply -2 by -m, which results in +2m, since multiplying two negative values gives a positive result.

When you've applied the distributive property here, the equation - 2(4-m) becomes - 8 + 2m. Understanding how to distribute terms is a vital skill for simplifying complex equations.

Rearranging Terms

Once you've distributed terms, the next step is to arrange similar terms together. This process is called rearranging. In the equation -8 + 2m = 10, our goal is to isolate the term with the variable from constants for easier solving. To do this:

• You would move the constant term -8 to the other side of the equation. You can accomplish solving this by performing the inverse operation. Add 8 to both sides, which cancels -8 on the left.
• The equation now looks like: 2m = 18.

Rearranging terms helps to simplify the equation, setting up the final steps to find the variable's value.

Isolating Variables

The main goal in solving an equation for a variable is to isolate it on one side, which allows you to find its value. With our equation: 2m = 18, the variable m is accompanied by the coefficient 2. To isolate m:- Divide both sides of the equation by 2. This step helps to eliminate the coefficient next to m, making it stand alone.Thus, when we divide: \[ m = \frac{18}{2} \]We solve this and find that m = 9.This technique is crucial as it systematically reduces the equation to its simplest form, making it possible to identify the variable's value.