Problem 49 - Elementary Algebra

Question

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction. $$ 12-(m-2)=2 m+3 m-2 m+3(5-3 m) $$

Step-by-Step Solution

Answer

Answer: The value of 'm' in the given equation is $$m = 1$$.

Step 1: Simplify the expressions on both sides of the equation

To simplify the given equation, $$12 - (m-2) = 2m + 3m - 2m + 3(5 - 3m)$$, start by simplifying the terms in the parentheses and distributing the coefficients as needed: $$12 - m + 2 = 2m + 3m - 2m + 15 - 9m$$

Step 2: Combine like terms

Now, combine like terms in the expressions on both sides of the equation: $$ - m + 14 = -4m + 17 $$

Step 3: Rearrange the equation

Next, we want to isolate 'm' on one side of the equation. To do this, add '4m' to both sides and subtract '14' from both sides: $$ - m + 14 + 4m = -4m + 17 + 4m - 14 $$ This simplifies to: $$ 3m = 3 $$

Step 4: Solve for 'm'

Now, divide both sides of the equation by '3' to solve for 'm': $$ \frac{3m}{3} = \frac{3}{3} $$ This simplifies to: $$ m = 1 $$ Since we've found a value for 'm', the given equation is a conditional equation, and the solution is $$m = 1$$.

Key Concepts

Algebraic Simplification

When tackling algebraic equations, it's crucial to first simplify expressions. Algebraic simplification involves breaking down complex expressions into simpler, more manageable parts. For the given equation, you start by examining any parentheses:

• Identify terms that need distribution, like \(3(5-3m)\).
• Execute the distribution by multiplying each term inside the parentheses by the number outside, resulting in \(15 - 9m\).

This process transforms the equation into a more straightforward form. During this stage, you'll work with combining and adjusting terms, ensuring that all calculations are accurate. This preparation lays the groundwork for the next steps in solving the equation.

Solving Equations

Once the expression is simplified, solving the equation involves finding the values of the variable that make the equation true. The key here is to isolate the variable (

• Start by seeing if the variable can be moved to one side of the equation using basic operations like addition or subtraction.
• In our example, rearranging terms involves adding \(4m\) to both sides, which gives \(3m = 3\).

After isolating the variable terms on one side, perform operations to solve for the variable. In our case, dividing both sides by \(3\) yields \(m = 1\). Solving equations requires careful handling of operations to maintain balance and achieve the correct answer.

Like Terms

Understanding like terms is fundamental for simplifying equations. Like terms in an equation have the same variable raised to the same power, thus they can be combined:

• In our example, terms like \(2m, 3m,\) and \(-2m\) are considered like terms because they all contain the variable \(m\).
• Similarly, constant terms like \(12\) and \(2\) can be combined separately.

By combining like terms, we simplify the structure of the equation. This process reduces the complexity, making it easier to manipulate and solve. Recognizing and combining like terms simplifies the mathematical process, paving the way for clearer solutions.