Problem 34 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 4 a+16=6 a+8 a+6 $$

Step-by-Step Solution

Answer

The solution is \( a = 1 \).

Step 1: Simplify the Equation

First, simplify the right-hand side of the equation. Combine the terms with variable 'a':\[ 6a + 8a + 6 = 14a + 6 \] Thus, the equation becomes:\[ 4a + 16 = 14a + 6 \]

Step 2: Eliminate Variable on One Side

To solve for 'a', we need to get all the 'a' terms on one side. Subtract \( 4a \) from both sides:\[ 16 = 14a - 4a + 6 \]This simplifies to:\[ 16 = 10a + 6 \]

Step 3: Isolate the Variable Term

Next, isolate the term with 'a' by subtracting 6 from both sides of the equation:\[ 16 - 6 = 10a \]Simplifying gives us:\[ 10 = 10a \]

Step 4: Solve for the Variable

Divide both sides by 10 to solve for 'a':\[ \frac{10}{10} = \frac{10a}{10} \]So, \( a = 1 \).

Step 5: Check the Solution

Substitute \( a = 1 \) back into the original equation to ensure it's correct:\[ 4(1) + 16 = 6(1) + 8(1) + 6 \]This simplifies to:\[ 4 + 16 = 6 + 8 + 6 \]\[ 20 = 20 \]The solution satisfies the original equation, confirming that \( a = 1 \) is correct.

Key Concepts

Equation Simplification

When solving linear equations, the first step often involves simplifying the equation. Simplification allows you to see the equation in its simplest form, making it easier to solve. Let's consider the given equation:
\[ 4a + 16 = 6a + 8a + 6 \]
We start by combining like terms on the right side of the equation. Like terms are terms that include the same variable raised to the same power. In this case, \( 6a \) and \( 8a \) can be combined because they both include the variable \( a \). When we add these together, we get \( 14a \).
Therefore, the equation simplifies to:
\[ 4a + 16 = 14a + 6 \]
Simplifying equations in this way can make it much easier to see what steps are needed next, paving the way for variable isolation.

Variable Isolation

After simplifying the equation, the next crucial step is to isolate the variable. Isolating the variable means rearranging the equation so that the variable you're solving for stands alone on one side. This often requires moving terms from one side of the equation to the other.
Continuing with our simplified equation:
\[ 4a + 16 = 14a + 6 \]
The goal is to get all the \( a \) terms on one side. We achieve this by subtracting \( 4a \) from both sides. This step ensures that the original balance of the equation stays intact:
\[ 16 = 14a - 4a + 6 \]
This simplifies to:
\[ 16 = 10a + 6 \]
Next, subtract 6 from both sides to further isolate the term with \( a \):
\[ 16 - 6 = 10a \]
Which simplifies to:
\[ 10 = 10a \]
Finally, divide both sides by 10 to solve for \( a \):
\[ a = 1 \]
By isolating the variable, we find our solution for \( a \). This process outlines the systematic approach needed to solve for unknowns in linear equations.

Checking Solutions

After finding a potential solution to our equation, checking your work is a vital part of the process. This step ensures the solution is accurate and that the value of the variable you found satisfies the original equation. Let's verify our solution by substituting \( a = 1 \) back into the original equation.
Original equation:
\[ 4a + 16 = 6a + 8a + 6 \]
Substitute \( a = 1 \):
\[ 4(1) + 16 = 6(1) + 8(1) + 6 \]
Simplifying both sides, we find:
\[ 4 + 16 = 6 + 8 + 6 \]
\[ 20 = 20 \]
Since both sides of the equation match when \( a = 1 \), this confirms the correctness of our solution. Checking solutions is important because it verifies the accuracy of the work done, ensuring any errors are caught before finalizing the answer.