Problem 54 - Algebra 2

Question

Solve each equation. Check your solution. $$ 4(k+3)+2=4.5(k+1) $$

Step-by-Step Solution

Answer

The solution is \(k = 19\).

Step 1: Expand the Equation

Distribute the numbers outside the parentheses on both sides of the equation. On the left side, distribute 4 into the terms inside the parentheses: \[4(k + 3) = 4k + 12\]On the right side, distribute 4.5 into the terms inside the parentheses:\[4.5(k+1) = 4.5k + 4.5\]

Step 2: Write the Expanded Equation

Substitute the expanded terms back into the equation:\[4k + 12 + 2 = 4.5k + 4.5\]

Step 3: Combine Like Terms

On the left side, combine the constant terms, 12 and 2:\[4k + 14 = 4.5k + 4.5\]

Step 4: Isolate the Variable Term

To isolate the variable term on one side, subtract 4k from both sides:\[14 = 0.5k + 4.5\]

Step 5: Solve for k

Subtract 4.5 from both sides to get:\[14 - 4.5 = 0.5k\] \[9.5 = 0.5k\]Now divide both sides by 0.5 to solve for \(k\):\[k = \frac{9.5}{0.5} = 19\]

Step 6: Check the Solution

Substitute \(k = 19\) back into the original equation to verify:Original equation: \[4(k+3) + 2 = 4.5(k+1)\]Substituting \(k = 19\): \[4(19+3) + 2 = 4.5(19+1)\]Calculate both sides: Left side: \[4 \times 22 + 2 = 88 + 2 = 90\]Right side: \[4.5 \times 20 = 90\]Since both sides are equal, \(k = 19\) is the correct solution.

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that allows us to multiply a single term by terms inside a set of parentheses. This method helps simplify expressions, and is especially useful when solving complex equations. In the equation \(4(k+3)+2=4.5(k+1)\), the distributive property is applied on both sides:

• On the left: \(4(k + 3)\) becomes \(4k + 12\).
• On the right: \(4.5(k + 1)\) becomes \(4.5k + 4.5\).

By distributing the numbers outside the parentheses, we simplify the equation into terms that can be easily combined and manipulated to find the variable. The distributive property thus transforms more complex expressions into a series of simpler terms.

Combining Like Terms

Once the distributive property has been applied, the next step involves combining like terms. Like terms are terms that have the same variable raised to the same power. In our example, after distributing, we have the expanded form:

• Left side: \(4k + 12 + 2\)
• Right side: \(4.5k + 4.5\)

On the left side, we see constants, \(12\) and \(2\), which can be combined into one term. This results in:

• \(4k + 14\) on the left side
• \(4.5k + 4.5\) on the right side

Combining like terms simplifies the equation, making it easier to isolate the variable, our next crucial step.

Isolating Variables

Isolating the variable is an essential part of solving an equation, allowing us to express the variable in terms of known quantities. To isolate \(k\) in our equation \(4k + 14 = 4.5k + 4.5\), we need to manipulate the equation by eliminating \(4k\) from both sides:

• Subtract \(4k\) from both sides to get \(14 = 0.5k + 4.5\).
• Then, eliminate the constant next to \(0.5k\) by subtracting \(4.5\) from both sides: \(14 - 4.5 = 0.5k\).
• This results in: \(9.5 = 0.5k\).

Finally, solve for \(k\) by dividing both sides by \(0.5\), resulting in \(k = 19\). This process of isolating the variable is key to finding its value.

Checking Solutions

After solving for the variable, checking the solution ensures that your calculations are correct and the variable satisfies the original equation. In our example, plugging \(k = 19\) back into the original equation:

• Left side: \(4(19+3) + 2\)
• Right side: \(4.5(19+1)\)

Calculate each side:

• Left side becomes \(4 \times 22 + 2 = 90\).
• Right side becomes \(4.5 \times 20 = 90\).

Since both sides equal \(90\), the solution \(k = 19\) is correct. Always checking the solution is a vital part of the process to ensure the accuracy of your result.