Solving Algebraic Expressions Find The Value Of 3ax-24by³+2026

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In this article, we delve into the fascinating world of algebraic expressions and equations. Our focus will be on solving a specific problem that involves evaluating an algebraic expression given certain conditions. We will explore the step-by-step process of substituting values, simplifying expressions, and ultimately finding the solution. This exploration will not only enhance your understanding of algebraic manipulations but also demonstrate the practical application of these concepts. Our primary goal is to provide a comprehensive guide that is both informative and engaging, ensuring that readers of all levels can grasp the underlying principles and techniques involved.

Problem Statement

The problem at hand presents us with an intriguing scenario. We are given that when x = 2 and y = -4, the expression ax³ + (1/2)by + 5 equals 7. Our challenge is to determine the value of the expression 3ax - 24by³ + 2026 when x = -4 and y = -1/2. This problem requires us to first find a relationship between the constants a and b using the initial condition and then apply this relationship to evaluate the second expression. The process involves careful substitution, algebraic manipulation, and a keen eye for detail. Let's embark on this journey together and unravel the solution.

Step-by-Step Solution

1. Substituting the Given Values

The initial step in solving this problem is to substitute the given values of x and y into the first equation. We are told that when x = 2 and y = -4, the expression ax³ + (1/2)by + 5 equals 7. This gives us a concrete starting point for our calculations. Substituting these values, we get:

a(2)³ + (1/2)b(-4) + 5 = 7

This substitution transforms the algebraic expression into a numerical equation involving a and b. This is a crucial step because it allows us to establish a relationship between these two constants. By simplifying this equation, we can isolate the terms involving a and b and pave the way for further analysis.

2. Simplifying the Equation

Now that we have substituted the values, the next step is to simplify the equation. This involves performing the arithmetic operations and rearranging the terms to make the equation more manageable. Let's break down the simplification process step by step.

First, we calculate 2³ which equals 8. So, the first term becomes 8a. Next, we multiply (1/2) by -4, which gives us -2. Thus, the second term becomes -2b. The equation now looks like this:

8a - 2b + 5 = 7

To further simplify, we subtract 5 from both sides of the equation. This isolates the terms involving a and b on one side and the constant term on the other side. This gives us:

8a - 2b = 2

This simplified equation is a significant milestone in our solution. It establishes a direct relationship between a and b, which we can use later to evaluate the second expression. To make this relationship even clearer, we can divide the entire equation by 2, which gives us:

4a - b = 1

This is our key equation, and we will refer back to it when we evaluate the second expression.

3. Expressing b in terms of a

To make our calculations easier in the next step, it is helpful to express one variable in terms of the other. In this case, we can easily express b in terms of a using the simplified equation we derived in the previous step. Our equation is:

4a - b = 1

To isolate b, we can add b to both sides and subtract 1 from both sides. This gives us:

4a - 1 = b

Now we have b expressed in terms of a. This is a crucial step because it allows us to substitute this expression for b in the second equation, effectively reducing the number of variables and making the evaluation process simpler. This substitution technique is a fundamental tool in algebra and is widely used in solving various types of problems.

4. Substituting the New Values of x and y

Now, let's shift our focus to the second expression, which we need to evaluate when x = -4 and y = -1/2. The expression is:

3ax - 24by³ + 2026

We substitute x = -4 and y = -1/2 into this expression. This gives us:

3a(-4) - 24b(-1/2)³ + 2026

This substitution is similar to what we did in the first step, but this time we are dealing with a different expression and different values of x and y. The key difference is that we now have an expression for b in terms of a, which we can use to further simplify this expression.

5. Simplifying the Second Expression

Now that we have substituted the new values of x and y, the next step is to simplify the second expression. This involves performing the arithmetic operations and substituting the expression for b that we found earlier. Let's break down this simplification process.

First, we calculate 3a(-4), which gives us -12a. Next, we calculate (-1/2)³, which equals -1/8. So, the second term becomes -24b(-1/8). Multiplying -24 by -1/8 gives us 3. Thus, the second term simplifies to 3b. The expression now looks like this:

-12a + 3b + 2026

Now, we substitute b = 4a - 1 into this expression. This gives us:

-12a + 3(4a - 1) + 2026

Next, we distribute the 3 across the terms inside the parentheses:

-12a + 12a - 3 + 2026

Notice that the -12a and +12a terms cancel each other out. This leaves us with:

-3 + 2026

6. Calculating the Final Value

The final step is to perform the remaining arithmetic operation to find the value of the expression. We have:

-3 + 2026

Adding these two numbers gives us:

2023

Therefore, the value of the expression 3ax - 24by³ + 2026 when x = -4 and y = -1/2 is 2023. This is our final answer.

Conclusion

In this article, we have walked through the step-by-step solution to an algebraic problem involving the evaluation of expressions given specific conditions. We started by substituting the given values into the first equation, simplifying the equation, and expressing one variable in terms of the other. Then, we substituted new values into the second expression, simplified it using the relationship we found earlier, and finally calculated the value of the expression. This problem demonstrates the power of algebraic manipulation and the importance of careful substitution and simplification. By following these steps, you can confidently tackle similar problems and enhance your understanding of algebra. The key takeaways from this exploration are the techniques of substitution, simplification, and expressing variables in terms of others. These are fundamental tools in algebra and will serve you well in solving a wide range of problems.