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Chapter 2: Problem 245

Solve each linear equation. $$-6+6(5-k)=15$$

Short Answer

Expert verified
k = 3/2

Step by step solution

01

- Distribute the constant

Distribute the 6 across the terms inside the parentheses: 6(5) - 6(k) becomes 30 - 6k. The equation now is: -6 + 30 - 6k = 15.
02

- Combine like terms

Combine the constant terms on the left-hand side: -6 + 30 = 24, so the equation is now: 24 - 6k = 15.
03

- Isolate the variable term

Subtract 24 from both sides to move the constant to the right: 24 - 6k - 24 = 15 - 24. This simplifies to: -6k = -9.
04

- Solve for k

Divide both sides by -6 to isolate the variable k: -6k / -6 = -9 / -6. ewline This simplifies to: k = 3/2.

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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 a fundamental concept in algebra. It allows us to multiply a term outside the parentheses with each term inside the parentheses. In our exercise, we start with the equation $$-6 + 6(5 - k) = 15$$.You distribute the 6 to both 5 and -k inside the parentheses.
This gives: $$6 \times 5 - 6 \times k$$, or simplified, $$30 - 6k$$. So the equation becomes: $$-6 + 30 - 6k = 15$$.
Distributive Property helps by breaking down complex expressions into simpler ones.
Combining Like Terms
Combining Like Terms is the process of simplifying algebraic expressions by merging terms that have the same variable raised to the same power. In our rearranged equation $$-6 + 30 - 6k = 15$$, the constants are -6 and 30.
By adding these constants together, we get $$24$$. Now the equation simplifies to $$24 - 6k = 15$$.
Combining like terms makes the equation easier to work with, by consolidating similar components.
Isolating the Variable
Isolating the Variable entails rearranging the equation to get the unknown variable on one side. Our equation now is $$24 - 6k = 15$$.
To isolate $$k$$, subtract $$24$$ from both sides: $$24 - 6k - 24 = 15 - 24$$.
This simplifies to: $$-6k = -9$$.
Isolating the variable helps in focusing on the unknown part of the equation, making it simpler to solve.
Simplifying Equations
Simplifying Equations involves performing arithmetic operations to reduce the equation to its simplest form. From $$-6k = -9$$, we divide both sides by $$-6$$ to solve for $$k$$.
This gives: $$k = \frac{-9}{-6}$$, which simplifies to: $$k = \frac{3}{2}$$.
Simplifying ensures the equation is manageable, and the solution is clear and precise.
Through these steps, we end with $$k = \frac{3}{2}$$, making the variable explicit.

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