Problem 33 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 3 k+6=5 k+10 $$

Step-by-Step Solution

Answer

The solution to the equation is \(k = -2\).

Step 1: Simplify the Equation

Start by subtracting \(3k\) from both sides of the equation to eliminate \(k\) from the left side: \[3k + 6 - 3k = 5k + 10 - 3k\]This simplifies to:\[6 = 2k + 10\]

Step 2: Isolate the Variable Term

Next, subtract 10 from both sides to isolate the variable term \(2k\):\[6 - 10 = 2k + 10 - 10\]Simplifies to:\[-4 = 2k\]

Step 3: Solve for the Variable

Divide both sides by 2 to solve for \(k\):\[\frac{-4}{2} = \frac{2k}{2}\]This gives:\[k = -2\]

Step 4: Verify the Solution

Substitute \(k = -2\) back into the original equation to check if it satisfies it:Original equation: \(3k + 6 = 5k + 10\)Substitute \(k = -2\):\(3(-2) + 6 = 5(-2) + 10\)Simplifies to:\(-6 + 6 = -10 + 10\)\(0 = 0\)Since both sides of the equation are equal, \(k = -2\) is a valid solution.

Key Concepts

Checking Solutions

Checking your solution is one of the most important steps when solving equations. It ensures that the value you found actually satisfies the original equation. This process involves substituting your solution back into the original equation to verify its correctness.
For instance, in the provided equation, after solving for \(k\), we obtained \(k = -2\). To check this solution, substitute \(-2\) back into the original equation:

• Original equation: \(3k + 6 = 5k + 10\)
• Substitute \(k = -2\): \(3(-2) + 6 = 5(-2) + 10\)
• Simplify both sides: \(-6 + 6 = -10 + 10\)

Both sides equal 0, confirming that our solution is correct. Always perform this check to avoid errors that can be easily overlooked during calculations.

Algebraic Manipulation

Algebraic manipulation is a fundamental technique used to simplify equations or expressions to make them solvable. This involves various operations such as adding, subtracting, multiplying, or dividing terms.
In the problem we addressed, we started by subtracting \(3k\) from both sides of the equation. This type of manipulation helps to move all terms involving the variable to one side, simplifying the equation:

• Equation: \(3k + 6 = 5k + 10\)
• Subtract \(3k\): \(6 = 2k + 10\)

It's crucial to perform the same operation on both sides to maintain the equivalence of the equation. Such steps help progressively bring the equation to a form where solving becomes straightforward.

Isolating Variables

Isolating the variable involves rearranging an equation to express the unknown variable on its own on one side of the equation. This is essential to solving for that variable effectively.
To isolate \(k\) in our example, we needed to eliminate any constants or coefficients connected with \(k\). After bringing the \(k\) terms to one side, the next step was:

• Equation after simplification: \(6 = 2k + 10\)
• Subtract 10 from both sides: \(-4 = 2k\)

Finally, divide both sides by 2 to isolate \(k\):

• \(k = \frac{-4}{2} = -2\)

The goal is to have \(k\) by itself, allowing us to identify the solution directly. This method is powerful in simplifying complex algebraic problems.