Problem 68 - Elementary Algebra

Question

Solve. $$ -12=-2 x+3 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{15}{2} \).

Step 1: Isolate the Variable Term

First, we need to get rid of the constant on the right-hand side. To do this, subtract 3 from both sides of the equation: \[-12 - 3 = -2x + 3 - 3\]This simplifies to: \[-15 = -2x\]

Step 2: Solve for the Variable

Next, to solve for \(x\), divide both sides by \(-2\):\[\frac{-15}{-2} = \frac{-2x}{-2}\]This gives:\[x = \frac{15}{2}\]

Key Concepts

Isolate the Variable

Isolating the variable is the first key step in solving linear equations. The goal is to rearrange the equation so that the variable you want to solve for is on one side of the equation, usually the right-hand side, leaving only constants on the other side. In the given equation \[-12 = -2x + 3\], isolating the variable means that we need to remove the constant term.

• To do this, subtract 3 from both sides of the equation.
• This action leaves the equation with only the variable term on one side.

So, the equation becomes \[-15 = -2x\].
By isolating the variable term, we're set up to solve for the variable in the next step.

Solve for x

Once the variable term is isolated, the next objective is to solve for that variable, in this case, \(x\). Solving for \(x\) involves manipulating the equation to get \(x\) alone on one side.

• In this problem, we have the equation \(-15 = -2x\).
• The coefficient of \(x\) is \(-2\).

The next move is to divide both sides of the equation by \(-2\) to eliminate the coefficient:
\[ \frac{-15}{-2} = x \]
Simplifying the fraction, we arrive at the solution \(x = \frac{15}{2} \). Thus, we have effectively solved the equation for \(x\), providing us with its value.

Step-by-Step Solution

Understanding a step-by-step solution is crucial in mastering the skill of solving linear equations. A detailed walkthrough of the solution helps identify essential processes and rationales. Let's break this down:

• The initial step in the provided solution is to isolate the variable by subtracting 3 from both sides. This removes extraneous numbers alongside \(x\) on the variable's side.
• Then, the equation simplifies to \(-15 = -2x\).
• Next, the solution needs dividing both sides by the coefficient \(-2\) to solve for \(x\).

These steps demonstrate a logical progression:

• Removing constants first,
• then addressing the multiplication by the coefficient.

By understanding this method, students can apply these steps to similar problems, aiding both comprehension and efficiency in math problem-solving.