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Chapter 2: Problem 64

Solve each equation, and check your solution. $$ 4(k-6)-(3 k+2)=-5 $$

Short Answer

Expert verified
k = 21. Check: 4(21-6)-(3(21)+2)=-5.

Step by step solution

01

- Distribute

Distribute the 4 to both terms inside the parenthesis on the left-hand side: 4(k-6)-(3k+2)=-54k - 24 - 3k - 2 = -5
02

- Combine Like Terms

Combine the terms with k and the constant terms on the left-hand side: 4k - 3k - 24 - 2 = -5 k - 26 = -5
03

- Isolate the Variable

Add 26 to both sides to isolate k: k - 26 + 26 = -5 + 26 k = 21
04

- Check the Solution

Substitute k = 21 back into the original equation to verify the solution: 4(21-6) - (3(21) + 2) = -5 4(15) - (63 + 2) = -5 60 - 65 = -5 -5 = -5, which is true. So, k = 21 is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is fundamental in algebra. It allows you to multiply a single term by each term within a parenthesis. In our exercise, we need to apply it to simplify the equation. When you have an equation like \( 4(k-6) - (3k + 2) = -5 \), distribute the 4 across the terms within the parentheses:
\( 4 \times k - 4 \times 6 = 4k - 24 \).
Next, apply the distributive property and simplify the expression, resulting in: \( 4k - 24 - 3k - 2 \).
Now, the equation looks simpler and gets us ready for the next step.
Combining Like Terms
Combining like terms helps to simplify the equation further. 'Like terms' are terms that have the same variable raised to the same power. In our case, look at \( 4k \) and \( -3k \) and the constants \( -24 \) and \( -2 \). Combining like terms simplifies the equation:
\( 4k - 3k = k \) and \( -24 - 2 = -26 \).
So the equation \( 4k - 24 - 3k - 2 = -5 \) becomes: \( k - 26 = -5 \).
This makes it easier to isolate the variable in the next step.
Isolating the Variable
To solve for the variable, you need to isolate it on one side of the equation. Our simplified equation now is \( k - 26 = -5 \). To isolate \( k \), add 26 to both sides:
\( k - 26 + 26 = -5 + 26 \).
This simplifies to \( k = 21 \).
Now, we know the value of \( k \) is 21. Ensuring that the variable is by itself on one side helps us find its value effectively.
Verifying Solutions
Verification is crucial as it confirms that our solution is correct. Substitute \( k = 21 \) back into the original equation to see if both sides are equal:
Original equation: \( 4(k-6) - (3k+2) = -5 \).
Substitute \( k = 21 \):
\( 4(21-6) - (3(21) + 2) = -5 \).
Simplify inside the parentheses first:
\( 4(15) - (63 + 2) = -5 \).
Then, perform the multiplication and addition/subtraction:
\( 60 - 65 = -5 \).
This results in \( -5 = -5 \), which is true. Therefore, our solution \( k = 21 \) is correct.

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