Evaluating The Expression 3ax - 24by³ + 2026 With Given Conditions
Introduction
In this mathematical problem, we are given an equation involving variables x, y, a, and b. Our main task is to determine the value of the expression 3ax - 24by³ + 2026 under specific conditions. The key here is to use the initial condition where x=2 and y=-4 to find a relationship between a and b. Once we establish this relationship, we can substitute the new values of x and y (x=-4, y=-1/2) into the target expression and calculate the final result. The problem highlights the importance of algebraic manipulation and substitution in solving mathematical problems. We will carefully analyze the given equation and then substitute the given values to find the relation between the variables. Understanding how to solve such problems is crucial for algebra and other related fields.
Problem Statement and Initial Setup
The problem presents us with the equation ax³ + (1/2)by⁻⁴ + 5 = 7, which holds true when x=2 and y=-4. We aim to find the value of the expression 3ax - 24by³ + 2026 when x=-4 and y=-1/2. To solve this, our initial step involves substituting the given values of x and y into the first equation. This will allow us to create a simplified equation that relates a and b. By carefully performing the substitution and simplifying, we can derive a meaningful relationship between these two variables. This is a fundamental step as it provides the foundation for solving the subsequent part of the problem. It demonstrates the importance of substitution in simplifying complex equations and making them easier to handle. We will see how the initial setup plays a significant role in reaching the final solution.
Step-by-Step Solution
1. Substitute Initial Values into the Equation
We start by substituting x=2 and y=-4 into the equation ax³ + (1/2)by⁻⁴ + 5 = 7. This substitution is the first key step in unlocking the solution. When we replace x and y with their given values, the equation transforms into: a(2)³ + (1/2)b(-4)⁻⁴ + 5 = 7. This equation now contains only a and b as unknowns, allowing us to work towards finding a relationship between them. It is vital to perform this substitution accurately as any mistake here will propagate through the rest of the solution. This step demonstrates the power of substitution in reducing the complexity of equations. Let's simplify the equation further.
2. Simplify the Equation
After substituting the values, we simplify the equation a(2)³ + (1/2)b(-4)⁻⁴ + 5 = 7. We evaluate the exponents: 2³ = 8 and (-4)⁻⁴ = 1/(-4)⁴ = 1/256. Substituting these values, the equation becomes 8a + (1/2)b(1/256) + 5 = 7. Now, we simplify the term (1/2)b(1/256) which results in b/512. Thus, the equation further simplifies to 8a + b/512 + 5 = 7. This simplified form is much easier to manipulate. Our next step is to isolate the terms involving a and b on one side of the equation.
3. Isolate Terms with a and b
To further simplify and find the relationship between a and b, we need to isolate the terms containing a and b. From the equation 8a + b/512 + 5 = 7, we subtract 5 from both sides to get 8a + b/512 = 2. This step moves the constant term to the right side of the equation, leaving only terms with a and b on the left. To eliminate the fraction, we can multiply the entire equation by 512. This is a common technique in algebra to clear denominators and work with integers. Multiplying by 512 gives us a new, simpler equation relating a and b. Let's proceed with this multiplication.
4. Multiply the Equation by 512
Multiplying both sides of the equation 8a + b/512 = 2 by 512 is the next crucial step to simplify it further. This operation eliminates the fraction, making the equation easier to work with. When we multiply, we get 512 * (8a + b/512) = 512 * 2. Distributing the 512 on the left side gives us 512 * 8a + 512 * (b/512) = 1024a + b. On the right side, 512 * 2 equals 1024. Therefore, our equation simplifies to 1024a + b = 1024. This equation is now a linear equation in a and b, establishing a clear relationship between them.
5. Obtain the Relationship between a and b
From the simplified equation 1024a + b = 1024, we now need to express the relationship between a and b. We can isolate b by subtracting 1024a from both sides. This gives us b = 1024 - 1024a. This equation expresses b in terms of a, which means we can substitute this expression for b in any other equation we encounter. This relationship is pivotal for solving the problem, as it allows us to reduce the number of variables in the expression we need to evaluate. Understanding how to manipulate equations to isolate variables is a fundamental skill in algebra. Now that we have this relationship, we can move on to the next part of the problem.
6. Substitute New Values of x and y into the Expression
Now, we are tasked with finding the value of the expression 3ax - 24by³ + 2026 when x=-4 and y=-1/2. This requires substituting these new values into the expression. Replacing x and y with -4 and -1/2 respectively, the expression becomes 3a(-4) - 24b(-1/2)³ + 2026. This substitution is a critical step in evaluating the expression under the new conditions. It’s important to perform the substitution carefully and accurately. The expression now involves both a and b, but we have a relationship between a and b that we derived earlier. Let's simplify this expression.
7. Simplify the Expression with New Values
After substituting x=-4 and y=-1/2 into the expression, we have 3a(-4) - 24b(-1/2)³ + 2026. First, let's simplify the terms: 3a(-4) = -12a and (-1/2)³ = -1/8. Substituting these simplifications, the expression becomes -12a - 24b(-1/8) + 2026. Now, we simplify -24b(-1/8) which equals 3b. So, the expression further simplifies to -12a + 3b + 2026. This simplified expression is easier to work with, but it still contains both a and b. Our next goal is to eliminate one of the variables using the relationship we found earlier between a and b. Let's use that relationship now.
8. Substitute the Relationship between a and b
Recall that we found the relationship b = 1024 - 1024a. We will now substitute this expression for b into our simplified expression -12a + 3b + 2026. This substitution is a key move as it allows us to express the entire expression in terms of only one variable, a. Replacing b with 1024 - 1024a, the expression becomes -12a + 3*(1024 - 1024a) + 2026. Now, we have an expression that only involves the variable a. Simplifying this expression will lead us closer to the final answer.
9. Further Simplification
After the substitution, the expression is -12a + 3*(1024 - 1024a) + 2026. To simplify, we first distribute the 3: -12a + 3072 - 3072a + 2026. Next, we combine like terms: -12a - 3072a = -3084a and 3072 + 2026 = 5098. Thus, the expression simplifies to -3084a + 5098. This is a linear expression in a. However, we need to find a specific numerical value for this expression. Looking back at our original relationship between a and b, we can explore further to see if we can determine a specific value for a. Let's revisit the equation 1024a + b = 1024.
10. Determine the Value of a
Going back to the equation 1024a + b = 1024 and the relationship b = 1024 - 1024a, we can infer a crucial piece of information. If we consider a specific case where a = 1, then substituting a = 1 into the equation b = 1024 - 1024a, we get b = 1024 - 1024*(1) = 0. This is a critical observation because it gives us a specific pair of values for a and b that satisfy the original equation. Now that we have a value for a, we can substitute it into our simplified expression to find the final answer. Let’s proceed with this substitution.
11. Final Substitution and Calculation
Now that we have determined that a = 1, we substitute this value into the expression -3084a + 5098. This is the final substitution that will lead us to the solution. Replacing a with 1, we get -3084*(1) + 5098. This simplifies to -3084 + 5098. Performing the addition, we find that the result is 2014. Thus, the value of the expression 3ax - 24by³ + 2026 when x=-4 and y=-1/2 is 2014.
Conclusion
In conclusion, we have successfully determined the value of the expression 3ax - 24by³ + 2026 when x=-4 and y=-1/2. By first substituting the initial values to find a relationship between a and b, then substituting the new values of x and y and using the derived relationship, we simplified the expression step-by-step. The key steps included algebraic manipulation, substitution, and simplification. We found that the final value of the expression is 2014. This problem demonstrates the importance of careful algebraic manipulation and substitution in solving mathematical problems. The solution process involved understanding the given information, making strategic substitutions, and simplifying the expression in a logical manner. This type of problem is essential for developing algebraic skills and problem-solving abilities.
