Problem 12 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. \(20=44-8(2-x)\)

Step-by-Step Solution

Answer

The solution to the equation \(20=44-8(2-x)\) is x = -1

Step 1: Simplify the equation

Simplify the equation by doing the arithmetic. Therefore, we distribute the '-8' to the '(2-x)'. This gives us: \(20 = 44 - 16 + 8x\)

Step 2: Re-arrange the equation

Rearrange the equation to isolate 'x'. Combine like terms on the right side giving us: \(20 = 28 + 8x\). Move the '28' over to the other side to isolate 'x' which gives us:\(-8 = 8x\)

Step 3: Solve for x

To solve for 'x', divide both sides of the equation by '8'. As a result, we get \(-1 = x\)

Step 4: Check the solution

Substitute the value of 'x' found back into the original equation. Thus we obtain, \(20=44-8(2-(-1)) = 44 - 8(3) = 44 - 24 = 20\). Thus confirming that the solution is correct.

Key Concepts

Equation Solving

The process of solving linear equations involves finding the value of the variable that makes the equation true. This journey starts by transforming the equation step-by-step until the unknown variable is isolated on one side of the equation. In the given example, the equation is:
\[20=44-8(2-x)\]
The first step in solving this equation is to simplify it. Simplification involves applying mathematical operations like distributing, combining like terms, and performing arithmetic operations.

• Distribute \(-8\) across the terms in the parenthesis \((2-x)\), to get \(20 = 44 - 16 + 8x\).
• Combine like terms to simplify further: \(20 = 28 + 8x\).

These steps are essential to transform a complex equation into a simpler one, making it manageable to solve efficiently.

Isolating the Variable

Isolating the variable is crucial in solving equations. This process involves rearranging the equation to have the variable on one side, often by using inverse operations. Returning to our example, after simplifying we have:
\[20 = 28 + 8x\]
Our goal here is to solve for \(x\) (the variable) and place it independently on one side. To do this:

• Subtract \(28\) from both sides of the equation to remove it from the right side: \(20 - 28 = 8x\).
• Which simplifies to: \(-8 = 8x\).
• Then, divide both sides by \(8\) to fully isolate \(x\), resulting in \(x = -1\).

Now, \(x\) stands alone, and we have the answer to our equation. Isolating the variable is about utilizing arithmetic operations to reorganize the equation conveniently.

Checking Solutions

Once you have a solution, it's essential to verify it. Verification assures that the solution satisfies the original equation. Let’s take the calculated result, \(x = -1\), and check if this correctly solves the original equation:
\[20 = 44 - 8(2-x)\]
To verify, replace \(x\) with \(-1\):

• First, compute inside the parentheses: \(2 - (-1) = 3\).
• Then, substitute back into the equation: \(20 = 44 - 8(3)\).
• Solve the operations: \(44 - 24 = 20\).

The left-hand side equals the right-hand side, confirming that the solution \(x = -1\) is indeed correct. Checking solutions is a surefire way to confirm accuracy and correctness in solving equations.