Solving Algebraic Equations Step-by-Step Guide
In the realm of mathematics, algebraic equations serve as fundamental tools for modeling and solving real-world problems. These equations establish relationships between variables and constants, and the process of finding the values of these variables that satisfy the equation is a core concept in algebra. This article delves into solving two distinct algebraic equations, providing a step-by-step approach to unravel the solutions. We will explore the techniques involved in simplifying equations, eliminating fractions, and isolating variables to arrive at the desired answers. Whether you're a student grappling with algebraic concepts or a seasoned mathematician seeking a refresher, this comprehensive guide will illuminate the path to solving these equations with clarity and precision.
Our journey begins with the equation (y+11)/6 + (y+1)/9 = (y+7)/4. This equation presents a challenge due to the presence of fractions. To overcome this hurdle, our initial step involves eliminating these fractions by finding the least common multiple (LCM) of the denominators. The denominators in this equation are 6, 9, and 4. The LCM of these numbers is 36. By multiplying both sides of the equation by 36, we effectively clear the fractions and transform the equation into a more manageable form. This transformation is a crucial step in simplifying the equation and paving the way for isolating the variable 'y'.
Step 1: Eliminate Fractions
To eliminate the fractions in the equation (y+11)/6 + (y+1)/9 = (y+7)/4, we need to find the least common multiple (LCM) of the denominators 6, 9, and 4. The LCM of these numbers is 36. Multiplying both sides of the equation by 36, we get:
36 * [(y+11)/6 + (y+1)/9] = 36 * [(y+7)/4]
This simplifies to:
6(y+11) + 4(y+1) = 9(y+7)
Step 2: Expand and Simplify
Next, we expand the expressions on both sides of the equation:
6y + 66 + 4y + 4 = 9y + 63
Combining like terms, we get:
10y + 70 = 9y + 63
Step 3: Isolate the Variable
To isolate the variable 'y', we subtract 9y from both sides:
10y - 9y + 70 = 9y - 9y + 63
This simplifies to:
y + 70 = 63
Now, subtract 70 from both sides:
y + 70 - 70 = 63 - 70
This gives us:
y = -7
Step 4: Verify the Solution
To ensure the accuracy of our solution, we substitute y = -7 back into the original equation:
((-7)+11)/6 + ((-7)+1)/9 = ((-7)+7)/4
Simplifying, we get:
4/6 + (-6)/9 = 0/4
2/3 - 2/3 = 0
0 = 0
Since the equation holds true, our solution y = -7 is correct.
Therefore, the value of y that satisfies the equation (y+11)/6 + (y+1)/9 = (y+7)/4 is -7. However, this option is not available in the given choices. Let's re-evaluate the steps to identify any potential errors.
Upon reviewing the steps, a mistake was identified in the final calculation. The correct solution should be:
y = -7
Thus, none of the provided options (a) -1, (b) 7, (c) 1, (d) -1/7 are correct. The correct answer is -7.
Now, let's tackle the second equation: (p+2)(p-3) + (p+3)(p-4) = p(2p-5). This equation involves products of binomials and requires careful expansion and simplification. Our strategy will involve expanding the products on both sides of the equation, combining like terms, and then isolating the variable 'p' to determine its value. This process highlights the importance of meticulous algebraic manipulation to arrive at the correct solution.
Step 1: Expand the Products
First, we expand the products on both sides of the equation:
(p+2)(p-3) = p^2 - 3p + 2p - 6 = p^2 - p - 6
(p+3)(p-4) = p^2 - 4p + 3p - 12 = p^2 - p - 12
p(2p-5) = 2p^2 - 5p
Substituting these back into the original equation, we get:
p^2 - p - 6 + p^2 - p - 12 = 2p^2 - 5p
Step 2: Simplify the Equation
Combining like terms on the left side of the equation, we have:
2p^2 - 2p - 18 = 2p^2 - 5p
Step 3: Isolate the Variable
To isolate the variable 'p', we first subtract 2p^2 from both sides:
2p^2 - 2p^2 - 2p - 18 = 2p^2 - 2p^2 - 5p
This simplifies to:
-2p - 18 = -5p
Next, we add 5p to both sides:
-2p + 5p - 18 = -5p + 5p
This gives us:
3p - 18 = 0
Now, add 18 to both sides:
3p - 18 + 18 = 0 + 18
This simplifies to:
3p = 18
Finally, divide both sides by 3:
3p / 3 = 18 / 3
This gives us:
p = 6
Step 4: Verify the Solution
To verify our solution, we substitute p = 6 back into the original equation:
(6+2)(6-3) + (6+3)(6-4) = 6(2(6)-5)
Simplifying, we get:
(8)(3) + (9)(2) = 6(12-5)
24 + 18 = 6(7)
42 = 42
Since the equation holds true, our solution p = 6 is correct.
Therefore, the solution of the equation (p+2)(p-3) + (p+3)(p-4) = p(2p-5) is p = 6.
In conclusion, we have successfully navigated the process of solving two distinct algebraic equations. The first equation, (y+11)/6 + (y+1)/9 = (y+7)/4, required us to eliminate fractions by finding the least common multiple of the denominators. After simplification and isolation of the variable 'y', we arrived at the solution y = -7. The second equation, (p+2)(p-3) + (p+3)(p-4) = p(2p-5), involved expanding products of binomials and combining like terms. Through careful algebraic manipulation, we determined the solution to be p = 6.
These examples underscore the importance of a systematic approach to solving algebraic equations. By breaking down complex equations into smaller, manageable steps, we can effectively unravel the solutions. The techniques employed in this article, such as eliminating fractions, expanding products, combining like terms, and isolating variables, are fundamental tools in the algebraist's toolkit. Mastering these techniques empowers us to tackle a wide range of algebraic problems with confidence and accuracy. Whether you are a student, educator, or simply an individual with a passion for mathematics, the knowledge gained from this exploration will undoubtedly enhance your problem-solving capabilities.
