Problem 37 - Elementary Algebra

Question

Solve each of the conditional equations. $$ m-12=0 $$

Step-by-Step Solution

Answer

Answer: The value of 'm' in the given linear equation is $$m=12$$.

Step 1: Identify the linear equation

The given conditional equation is: $$m-12=0$$

Step 2: Isolate the variable 'm'

To isolate 'm', add 12 to both sides of the equation: $$m-12+12=0+12$$

Step 3: Simplify the equation

Simplify the equation by adding the numbers: $$m=12$$

Step 4: Check the solution

Substitute the value of 'm' back into the original equation to check if it holds true: $$m-12=0$$ $$12-12=0$$ $$0=0$$ Since the original equation holds true, the solution $$m=12$$ is correct.

Key Concepts

Conditional Equations

Conditional equations are mathematical statements that are true only under certain conditions. These conditions are the values that satisfy the equation. In the context of variable manipulation, a conditional equation often has one variable and will only be true when that variable has a specific value. For example, in the equation \( m-12=0 \), the condition that makes this equation true is when \( m \) equals 12. Unlike identities, which are true for all variable values, conditional equations are like puzzles waiting to be solved for that special number that makes them true.

Isolate the Variable

To solve a conditional equation, the goal is to 'isolate the variable'—this means to get the variable by itself on one side of the equation. The purpose of this is to find the value of the variable that will satisfy the conditional equation. In our example, to isolate \( m \), we need to eliminate everything else from its side of the equation. By adding 12 to both sides of \( m-12=0 \), we balance the equation and achieve \( m=12 \), where m is isolated. It's essential to use inverse operations—like adding the opposite number or multiplying by the reciprocal—to keep the equation balanced as we isolate the variable.

Equation Simplification

Once we've added 12 to both sides of the equation, we're left with \( m-12+12=0+12 \). The next crucial step is equation simplification, which involves combining like terms and making the equation as clean and straightforward as possible. Here, we simply add the numbers together to get \( m=12 \). Simplification makes the equation easier to understand and highlights the isolated variable's value, ensuring that you have a clear path to the solution. Always remember that simplification should maintain the equality of both sides of the equation.

Solution Checking

The last and critical step in solving conditional equations is to check that your solution is correct. This is done by substituting the value of the isolated variable back into the original equation to see if the equation holds true. For the equation \( m-12=0 \) and our solved value of \( m=12 \), we replace \( m \) with 12 and get \( 12-12=0 \), which simplifies to \( 0=0 \). A true statement confirms that we've found the right value. Checking your solution is a good habit because it can help catch any errors made during the problem-solving process.