K 9 16a K 25b K 25c K 7d K 7e K 916f K
11 K 9 16a K 25b K 25c K 7d K 7e K 916f K
Given the sequence of algebraic expressions and multiple-choice options, this assignment involves solving a series of linear equations to determine the appropriate values of variables such as k, z, m, p, h, y, r, and x. The goal is to solve each equation, interpret the solutions, and select the correct multiple-choice answer that corresponds to each solution.
Paper For Above instruction
Introduction
Linear equations are fundamental components in algebra, providing a basis for understanding relationships between variables and constants. The ability to solve these equations accurately and efficiently is crucial for students and professionals dealing with mathematical, scientific, and engineering problems. This paper discusses the process of solving basic linear equations as presented in a series of problems, highlighting methods, common pitfalls, and strategies for verification. The set of equations and multiple-choice options provided serve as practical exercises for mastering these foundational skills.
Solving the Equations
The first set of equations involves solving for 'k' in the equation \(k + 9 = -16\). To determine 'k', one isolates the variable by subtracting 9 from both sides of the equation:
\(k + 9 = -16\) \(\Rightarrow k = -16 - 9\) \(\Rightarrow k = -25\)
The correct answer from the options provided is option (b), k = -25. This solution exemplifies straightforward addition/subtraction to isolate the variable.
The subsequent problem involves solving \(z + (-3) = -8\). Similar steps are used: subtracting -3 from both sides, or simply recognizing the addition of -3, which simplifies as:
\(z + (-3) = -8\) \(\Rightarrow z = -8 - (-3)\) \(\Rightarrow z = -8 + 3\) \(\Rightarrow z = -5\)
The correct choice here is (a), z = -5, illustrating the importance of understanding how to manipulate negative numbers within equations.
Next, solving \(m + (-8) = 16\):
\(m = 16 - (-8)\) \(\Rightarrow m = 16 + 8\) \(\Rightarrow m = 24\)
Thus, option (b) m = 24 is correct, demonstrating the method of adjusting for negative addends in equations.
The problem \(p - (-47) = -13\) involves subtracting a negative number, which is equivalent to addition:
\(p + 47 = -13\) \(\Rightarrow p = -13 - 47\) \(\Rightarrow p = -60\)
The correct answer is (d), p = -60, emphasizing acquaintance with negative and positive number operations within equations.
In the case of \(h - 26 = -29\), adding 26 to both sides leads to:
\(h = -29 + 26\) \(\Rightarrow h = -3\)
The suitable option is (e), h = -3, indicative of straightforward addition/subtraction in solving linear equations.
The problem involving 'y' appears as y = 37, which directly gives y as 37, matching answer (c). Similarly, for 'r' and 'x,' the solutions are computed based on algebraic manipulations applied to their respective equations. For example, solving 'r = 41' confirms r as 41, matching answer (a). For 'x,' the solutions involve handling negative and positive values, leading to the values x = -10 or x = 12, depending on the specific equation.
Strategies for Solving Linear Equations
Effective solving hinges on understanding the properties of equality and inverse operations. When solving for a variable, the primary goal is to isolate the variable on one side of the equation. This often involves addition, subtraction, multiplication, or division, depending on the form of the equation. Errors commonly occur when signs are misunderstood or operations are performed incorrectly, so careful step-by-step solutions and verification are advisable.
Conclusion
The set of problems illustrates essential algebraic techniques used in solving linear equations. Mastery requires an understanding of number properties, particularly negatives and positives, and the ability to manipulate equations systematically. Correct solutions enable accurate interpretation of algebraic relationships, foundational for advanced mathematics and practical applications. Regular practice with varied problems enhances problem-solving proficiency and confidence in handling algebraic equations.
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