Henry's Detailed Solution To The Equation 4(-1/2x + 7) + 5x = 15

Understanding the Problem:

In this article, we will meticulously analyze Henry's step-by-step solution to the algebraic equation $4(-\frac{1}{2}x + 7) + 5x = 15$. Our aim is to not only understand each step but also to highlight the underlying mathematical principles and properties that govern the simplification and solution of such equations. We will delve into the distributive property, combining like terms, and the application of inverse operations to isolate the variable. This detailed examination will serve as a valuable resource for anyone seeking to enhance their understanding of algebraic problem-solving. Algebra, as a fundamental branch of mathematics, provides the tools and techniques to model and solve real-world problems. Equations, in particular, form the cornerstone of algebraic thinking, allowing us to represent relationships between variables and constants. Mastering the art of solving equations is crucial for success in higher-level mathematics and various scientific disciplines.

Step 1: Applying the Distributive Property

Henry's initial step involves applying the distributive property to the term $4(-\frac{1}{2}x + 7)$. The distributive property, a cornerstone of algebraic manipulation, dictates that $a(b + c) = ab + ac$. In this context, $a = 4$, $b = -\frac{1}{2}x$, and $c = 7$. Applying this property, we multiply 4 by both terms inside the parentheses:

$4 * (-\frac{1}{2}x) = -2x$ $4 * 7 = 28$

Thus, the expression $4(-\frac{1}{2}x + 7)$ simplifies to $-2x + 28$. The equation now becomes $-2x + 28 + 5x = 15$. This step is critical as it eliminates the parentheses, paving the way for combining like terms and further simplification. A common mistake here is to only multiply 4 by the first term inside the parentheses, neglecting the second term. Another error could arise from incorrect multiplication, especially with negative numbers and fractions. Therefore, meticulous application of the distributive property is paramount.

Step 2: Combining Like Terms

Following the distributive property, Henry proceeds to combine like terms. Like terms are terms that contain the same variable raised to the same power. In the expression $-2x + 28 + 5x = 15$, the like terms are $-2x$ and $5x$. Combining them involves adding their coefficients:$-2x + 5x = (-2 + 5)x = 3x$

The equation now transforms into $3x + 28 = 15$. This simplification reduces the number of terms, making the equation easier to solve. It's crucial to correctly identify like terms and perform the addition or subtraction accurately. A frequent error is to combine terms that are not alike, such as adding $3x$ and $28$. The constant term, 28, cannot be combined with the term containing the variable, $3x$. This step showcases the importance of understanding the structure of algebraic expressions and the rules governing their manipulation. By combining like terms, we are essentially simplifying the equation while preserving its equality.

Step 3: Isolating the Variable Term

To isolate the term containing the variable, $3x$, Henry subtracts 28 from both sides of the equation. This step utilizes the subtraction property of equality, which states that if $a = b$, then $a - c = b - c$. By subtracting 28 from both sides, we maintain the balance of the equation while effectively moving the constant term to the right side: $3x + 28 - 28 = 15 - 28$ leading to $3x = -13$

This step is crucial because it brings us closer to isolating the variable, $x$. The choice of subtracting 28 is strategic, as it cancels out the +28 on the left side, leaving only the term with the variable. A common mistake is to add 28 to both sides, which would further complicate the equation. Another potential error is performing the subtraction incorrectly, especially with negative numbers. It’s essential to remember that performing the same operation on both sides of the equation maintains equality. Subtraction Property of Equality helps us to simplify the equation and continue our journey towards finding the value of $x$. Understanding and applying this principle accurately is a vital skill in algebraic problem-solving. The result of this step, $3x = -13$, sets the stage for the final step, where we isolate the variable itself.

Step 4: Solving for x

In the final step, Henry isolates $x$ by dividing both sides of the equation by 3. This step utilizes the division property of equality, which asserts that if $a = b$, then $\frac{a}{c} = \frac{b}{c}$ (provided $c \neq 0$). Dividing both sides of $3x = -13$ by 3, we get: $\frac{3x}{3} = \frac{-13}{3}$ leading to $x = -\frac{13}{3}$

This final operation reveals the solution to the equation. The variable $x$ is now isolated, and its value is determined to be $-\frac{13}{3}$. It's crucial to divide both sides by the coefficient of $x$ to completely isolate the variable. A common error here is to multiply both sides by 3 instead of dividing, which would lead to an incorrect solution. Another mistake could be mishandling the negative sign, especially when dividing negative numbers. This step demonstrates the power of inverse operations in solving equations. By applying the inverse operation of multiplication (division), we undo the multiplication by 3, effectively isolating $x$. The final answer, $x = -\frac{13}{3}$, represents the value that, when substituted back into the original equation, will satisfy the equality. This step concludes the solution process, but it's always a good practice to check the solution by substituting it back into the original equation to ensure accuracy.

Verification of the Solution

To ensure the correctness of the solution, we substitute $x = -\frac{13}{3}$ back into the original equation:

$4(-\frac{1}{2}x + 7) + 5x = 15$

Substitute $x = -\frac{13}{3}$:

$4(-\frac{1}{2}(-\frac{13}{3}) + 7) + 5(-\frac{13}{3}) = 15$

Simplify the expression:

$4(\frac{13}{6} + 7) - \frac{65}{3} = 15$

Find a common denominator for the terms inside the parentheses:

$4(\frac{13}{6} + \frac{42}{6}) - \frac{65}{3} = 15$

Add the fractions inside the parentheses:

$4(\frac{55}{6}) - \frac{65}{3} = 15$

Multiply:

$\frac{220}{6} - \frac{65}{3} = 15$

Simplify the first fraction:

$\frac{110}{3} - \frac{65}{3} = 15$

Subtract the fractions:

$\frac{45}{3} = 15$

Simplify:

$15 = 15$

The left side of the equation equals the right side, confirming that our solution, $x = -\frac{13}{3}$, is correct. This verification step is crucial as it serves as a safeguard against potential errors made during the solving process. By substituting the solution back into the original equation and simplifying, we can ascertain whether the equality holds true. If the two sides of the equation are equal, the solution is valid; otherwise, an error has occurred and needs to be identified and corrected. The process of verifying the solution reinforces understanding and builds confidence in the problem-solving skills.

Conclusion:

Henry's meticulous step-by-step solution provides a clear roadmap for solving linear equations. By carefully applying the distributive property, combining like terms, and utilizing inverse operations, he successfully isolated the variable and found the solution. The verification step further solidifies the correctness of the solution, highlighting the importance of accuracy and thoroughness in algebraic problem-solving. This detailed analysis underscores the fundamental principles of algebra and serves as a valuable learning resource for anyone seeking to master the art of solving equations. The ability to solve equations is a fundamental skill that extends far beyond the classroom, finding applications in various fields such as science, engineering, economics, and computer science. By understanding the underlying concepts and practicing regularly, one can develop proficiency in solving equations and unlock the power of algebraic thinking. Remember, mathematics is a journey, and each solved equation is a step forward in that journey.