Problem 138 - College Algebra

Question

This will help you prepare for the material covered in the next section. $$\text { Solve: }-2 x-4-x+5$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1/3\).

Step 1: Combine like terms

Combine the 'x' terms together and the constants terms separately. This will lead to the formation of a simplified equation, which is \(-2x - x = 4 - 5\).

Step 2: Simplify the equation

\(-2x - x\) gives \(-3x\), and \(4 - 5\) gives \(-1\). Hence, the simplified equation is \(-3x = -1\).

Step 3: Solve for x

To isolate x, divide both sides of the equation by -3, this will make x the subject of the equation. Hence, \(x = -1 / -3\), which simplifies to \(x = 1 / 3\).

Key Concepts

Combining Like Terms

Understanding how to combine like terms is important for simplifying algebraic expressions. Like terms are terms that contain the same variables raised to the same power. Only the coefficients of these terms can differ. When you are dealing with a linear equation, like the one in our example \( -2x - 4 - x + 5 \), you'll notice that \( -2x \) and \( -x \) are like terms because they are both coefficients of \( x \) without any exponents.

Let's break it down: We have two terms that involve the variable \( x \) (\( -2x \) and \( -x \) ) and two constant terms (\( -4 \) and \( +5 \) ). To combine these, you simply add or subtract the coefficients of like terms. Here, \( -2x \) and \( -x \) become \( -3x \) when combined because \( -2 - 1 = -3 \). For the constants, \( -4 \) and \( +5 \) combine to \( +1 \) since \( -4 + 5 = 1 \). The knowledge of combining like terms directly influences the ease of simplifying any equation.

Simplifying Equations

Simplifying equations involves reducing them to their simplest form, making them easier to solve. This process includes combining like terms, as discussed previously, and performing basic arithmetic operations to condense the equation. For instance, once we've combined like terms in our example, we proceed by performing the simple subtraction between the constants.

\textbf{Performing the Arithmetic:} From \( -3x \) on the left and \( 1 \) on the right, you can't simplify any further because there are no more like terms to combine. It's critical at this point to ensure every term is in its simplest form before moving to solve for the variable. Remember not to rush; double-check each step to avoid simple mistakes that could lead to the wrong solution.

Isolate the Variable

The core aim of solving linear equations is to isolate the variable. This means getting the variable alone on one side of the equation. Directly following the simplification, our equation looks like \( -3x = -1 \).

To isolate \( x \) in this equation, we need to undo the multiplication of \( x \) by \( -3 \). How do we do that? By doing the opposite: dividing both sides of the equation by \( -3 \). This leaves us with \( x = (-1) / (-3) \).

Why does this work? Division is the inverse operation of multiplication. Thus, when we divide by the same number we MULTIPLIED by, we effectively cancel out the operation, leaving \( x \) all by itself, which is the ultimate goal. Now we have our solution, \( x = 1/3 \), a neat, isolated variable.