Problem 35 - Elementary Algebra

Question

Solve. $$ 6 x+2-3 x=-2 x-13 $$

Step-by-Step Solution

Answer

The solution is \(x = -3\).

Step 1: Simplify Both Sides

Start by simplifying both sides of the equation. On the left side, combine like terms: \(6x - 3x + 2 = 3x + 2\). The right side remains \(-2x - 13\). The equation now reads: \(3x + 2 = -2x - 13\).

Step 2: Eliminate the Variable on One Side

To remove the variable from one side, add \(2x\) to both sides: \(3x + 2x + 2 = -2x + 2x - 13\). This simplifies to \(5x + 2 = -13\).

Step 3: Isolate the Variable Term

Isolate the term with \(x\) by subtracting \(2\) from both sides: \(5x + 2 - 2 = -13 - 2\). This simplifies to \(5x = -15\).

Step 4: Solve for x

Finally, solve for \(x\) by dividing both sides by 5: \(x = \frac{-15}{5}\). This gives \(x = -3\).

Key Concepts

Combining Like Terms

When working with linear equations, one of the first tasks is to simplify the expressions by combining like terms. Like terms are terms that have the same variable raised to the same power. In our example, consider the equation:\[6x + 2 - 3x = -2x - 13\]On the left side, both \(6x\) and \(-3x\) are like terms because they both have the variable \(x\) with an exponent of one. By adding their coefficients together, \(6 - 3\), we simplify the expression to \(3x\). This step is known as combining like terms. The result transforms the left side into: \[3x + 2\] By combining like terms, you make the equation more straightforward, which simplifies further steps in solving the equation.

Isolating the Variable

Once the equation is simplified, the next step is to isolate the variable. This means getting the variable by itself on one side of the equation so that we can solve for it. After combining terms, we have:\[3x + 2 = -2x - 13\]The goal is to have \(x\) on one side only. To isolate \(x\), we can add \(2x\) to both sides, which eliminates the \(-2x\) from the right side and groups the \(x\) terms on the left side:\[3x + 2x + 2 = -13\]This simplifies to: \[5x + 2 = -13\] Now, subtract \(2\) from both sides to further isolate terms involving \(x\):\[5x = -15\] The variable \(x\) is now isolated, which means we are ready to solve for it.

Inverse Operations

The principle of inverse operations is fundamental in solving equations. It involves using opposite operations to 'undo' each step of the arithmetic and isolate the variable. Here, with the new expression:\[5x = -15\]We need to solve for \(x\). This requires using inverse operations. Since \(5x\) means \(5 \cdot x\) or "5 times \(x\)", the inverse operation here is division.To isolate \(x\), divide both sides by 5:\[x = \frac{-15}{5}\]Calculating the fraction yields:\[x = -3\] By applying inverse operations correctly, we solve the equation. Inverse operations ensure every manipulation remains algebraically valid, leading us to the correct value of the variable.