Problem 6 - Prealgebra

Question

Solve equation. Check your solution. \(4 k+24=6 k-10\)

Step-by-Step Solution

Answer

The solution is \(k = 17\). Verified by substitution.

Step 1: Set the Equation

The given equation is \(4k + 24 = 6k - 10\). Our goal is to isolate \(k\) and solve for its value.

Step 2: Rearrange Terms

Move the terms involving \(k\) to one side and constant terms to the other side. Subtract \(4k\) from both sides: \[ 24 = 6k - 4k - 10 \]This simplifies to: \[ 24 = 2k - 10 \]

Step 3: Isolate k

Add 10 to both sides of the equation to remove the constant on the right side: \[ 24 + 10 = 2k \]This simplifies to: \[ 34 = 2k \]

Step 4: Solve for k

Divide both sides by 2 to solve for \(k\): \[ k = \frac{34}{2} \]This simplifies to: \[ k = 17 \]

Step 5: Check Your Solution

Substitute \(k = 17\) back into the original equation to verify the solution: Original equation: \[ 4(17) + 24 = 6(17) - 10 \] Calculate each side: - Left side: \(4 \times 17 + 24 = 68 + 24 = 92\)- Right side: \(6 \times 17 - 10 = 102 - 10 = 92\)Since both sides are equal, \(k = 17\) is the correct solution.

Key Concepts

Understanding Prealgebra

Prealgebra is the foundation of all algebra concepts and helps students transition from basic arithmetic to more complex math. It introduces variables, expressions, and simple equations, laying the groundwork for future math learning.
A key skill in prealgebra is learning how to work with variables. Variables are symbols, often letters, that represent numbers in an equation.
For example, in the equation \(4k + 24 = 6k - 10\), "\(k\)" is a variable. Understanding how to use and manipulate variables is at the heart of prealgebra.The following steps are common in prealgebra when solving equations:

• Identify terms and constants: Terms are either variables or numbers connected by addition or subtraction. Constants are standalone numbers.
• Use inverse operations to solve equations: Inverse operations help in simplifying an equation by reversing operations. For instance, subtraction is the inverse of addition.

The goal of prealgebra is to make you comfortable with solving equations using these principles.

Introduction to Linear Equations

Linear equations are fundamental in algebra. They describe a straight line when graphed and have one or more variables raised only to the first power.
A linear equation has the standard form \(ax + b = c\), where \(a\), \(b\), and \(c\) are numbers, and \(x\) is the variable. This equation represents a balance, and solving it means finding the value of \(x\) that keeps both sides equal.Linear equations follow predictable steps for solving:

• Combine like terms on each side of the equation.
• Use inverse operations to get the variable on one side of the equation and the constants on the other side.

In our original exercise \(4k + 24 = 6k - 10\), you can see these principles in action. We simplified and rearranged the equation to isolate \(k\), eventually solving for its value. Linear equations like these are everywhere in math and used to model numerous real-world scenarios.

Mastering Algebraic Manipulation

Algebraic manipulation is the art of rearranging and simplifying equations to find solutions. It's a crucial skill in math, allowing you to see the relationships between different parts of an equation.
In our given equation \(4k + 24 = 6k - 10\), the key steps involved:

• Rearranging terms: Moving terms across the equation changes their signs, which is a basic component of simplifying equations.
• Using inverse operations: Subtracting, adding, multiplying, or dividing both sides of the equation by the same number is a way to simplify and solve for the variable.

For instance, subtracting \(4k\) from both sides was a strategic move to simplify the equation from \(4k + 24 = 6k - 10\) to \(24 = 2k - 10\).
Mastering this skill means being able to adjust and reframe equations until they are easy to work with. It's essential for solving not just linear equations but also more complex ones you might encounter later in math.