Problem 12 - Introductory Algebra for College Students

Question

Solve each 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 is \(x = -1\).

Step 1: Simplify the Equation

The first step is to simplify the equation so that there isn't a multiplication within parentheses on the right side. This requires distributing the '-8' across the '(2-x)'. This results in: \(20 = 44 - 8*2 + 8x\). Simplify this further to: \(20 = 44 - 16 + 8x\), which equates to: \(20 = 28 + 8x\).

Step 2: Isolate the variable

The next step is to isolate the variable 'x'. To do this, you must subtract '28' from both sides of the equation to keep it equivalent. Doing so results in: \(-8 = 8x\). Then divide both sides by '8' which gives: \(-1 = x\).

Step 3: Check the solution

Lastly, substitute \(-1\) back into the original equation to verify that it holds true. This means replacing every instance of 'x' in \(44-8(2-x)\) with \(-1\). The equation becomes: \(44-8(2-(-1))\), equals to, \(44-8(2+1)\), comprehensive turns into: \(44-8(3)\), which simplified provides: \(44-24 = 20\). Hence, \(-1\) is indeed the correct solution.

Key Concepts

The Distributive Property in Equations

The distributive property is a fundamental concept in mathematics that allows us to simplify equations. It involves distributing or multiplying a single term through a sum or difference within parentheses. In the given problem, we apply the distributive property to eliminate the parentheses on the right side of the equation. The original equation is \(20 = 44 - 8(2-x)\).
Here, we distribute the \(-8\) across the terms inside the parentheses:

• Multiply \(-8\) by \(2\): which becomes \(-16\).
• Multiply \(-8\) by \(-x\): which gives us \(+8x\).

After applying these operations, the equation becomes: \(20 = 44 - 16 + 8x\).
By using the distributive property, we convert a more complex equation into a simpler, linear one. This step is crucial because it helps us to easily work with and manipulate the equation in subsequent steps.

Isolating the Variable

When solving for a variable, our goal is to isolate it on one side of the equation. This process allows us to determine the value of the variable. Here, our task is to solve for \(x\) in the equation: \(20 = 28 + 8x\).
To isolate \(x\), we need to move all other terms to the opposite side of the equation. Here's how we do it:

• Subtract \(28\) from both sides: \(20 - 28 = 8x\).
• Simplify the left side: \(-8 = 8x\).

The next step is to get \(x\) by itself by dividing both sides by \(8\):

• Divide \(-8\) by \(8\): which results in \(x = -1\).

By performing these operations, we successfully isolate \(x\) and discover its value. Isolating the variable ensures we find the correct solution to the equation.

Checking Solutions for Accuracy

After finding a solution, it's essential to verify its correctness by checking if it satisfies the original equation. This final step ensures that the solution is accurate and valid. For the problem at hand, we found \(x = -1\).
To check the solution, we substitute \(-1\) back into the original equation: \(20 = 44 - 8(2-x)\). Here's the step-by-step check:

• Replace \(x\) with \(-1\): \(44 - 8(2-(-1))\).
• Calculate inside the parentheses: \(2 - (-1) = 2 + 1 = 3\).
• Proceed with the multiplication: \(-8 \times 3 = -24\).
• Complete the calculation: \(44 - 24 = 20\).

Since the left side equals the right side after substitution, our solution is confirmed to be correct. Checking solutions helps prevent errors and confirms that every step in the solving process was executed correctly.