Problem 131 - Introductory Algebra for College Students

Question

Solve and check: \(5 x+16=3(x+8)\).

Step-by-Step Solution

Answer

The solution for 'x' in this equation is 4.

Step 1: Distribute the right side

Distribute 3 on \(x+8\), thus the equation becomes \(5x + 16 = 3x + 24\).

Step 2: Equisate the variable 'x' on both sides

Subtract \(3x\) from each side of the equation in order to collect all terms with 'x' on one side and the constants on the other side. The equation now is \(2x + 16 = 24\).

Step 3: Isolate the variable

Subtract 16 from each side to isolate 'x'. This results in \(2x = 8\).

Step 4: Solve for 'x'

Finally, divide both sides by 2 to solve for 'x', which gives 'x = 4'.

Step 5: Check solution

Substitute the solution into the original equation: \(5*4 + 16 = 3*(4 + 8)\). After the calculation, we can see that both sides of the equation equal 36, hence the solution is correct.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions by distributing (or spreading out) multiplication over addition or subtraction within parentheses. This property states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. In the given problem, we apply the distributive property to \(3(x + 8)\).

• Here, 3 is multiplied by both \(x\) and 8 separately.
• This transforms the expression to \(3x + 24\).

Recognizing when and how to use the distributive property can simplify complex equations, setting up a more straightforward path to solve for the unknown variable.

Isolating Variables for Solution

Isolating variables is a critical step in solving equations wherein we try to have the variable on one side of the equation and constants on the other. Think of it like organizing—put all the similar items together. In our example, we start with the equation \(5x + 16 = 3x + 24\).
The goal is to get 'x' alone on one side:

• Subtract \(3x\) from both sides, giving you \(2x + 16 = 24\).
• Then, subtract 16 from both sides to further isolate the term with 'x', resulting in \(2x = 8\).

By isolating 'x', we simplify the equation, making it easier to solve for the unknown value.

Checking Your Solutions

Once you've solved an equation, it's crucial to verify the solution. This ensures that there are no mistakes in your calculations and that you've found the correct answer. To check the solution, substitute the found value back into the original equation. For the equation \(5x + 16 = 3(x + 8)\), we calculate as follows:

• Replace 'x' with 4 in both sides: \(5 \times 4 + 16\) and \(3 \times (4 + 8)\).
• Calculate both sides: \(20 + 16\) and \(3 \times 12\).
• Both sides equal 36, which confirms the solution \(x = 4\) is correct.

Checking solutions is a vital practice, as it not only confirms the correctness of your answer but also reinforces the solution process itself, enhancing understanding and accuracy.