Solving Equations: Find The Value Of An Expression
Hey math enthusiasts! Today, we're diving into a fun problem that combines algebra and a bit of logical thinking. The task is pretty straightforward: we're given a set of equations and some conditions, and our goal is to find the value of a specific expression. Sounds cool, right? Let's break it down step by step and see how we can crack this math nut. We'll be using some basic algebraic principles, so if you're a bit rusty, don't worry – I'll guide you through it.
Understanding the Problem: The Foundation of Our Solution
Alright, guys, let's get our heads around the core of the problem. We've got a system of equations, and each equation provides us with a relationship between different variables: a, b, c, and d. The problem gives us the following equations:
a + 3b = 73b + 2c = 9c + d = 1
Also, a, b, c, and d are all natural numbers (N*), meaning they are positive whole numbers. This is a crucial piece of information. It limits the possible values these variables can take. Why is this important? Because it helps us narrow down the solutions and find the right answers. We aren't just looking for any numbers that satisfy the equations; we need natural numbers. This constraint makes the problem much more manageable and allows us to use some clever tricks to find the values.
Our ultimate goal is to figure out the value of the expression 2a + 15b + 10c + 4d. To do this, we need to find the values of a, b, c, and d first. Think of it like a puzzle. We have several clues (the equations), and we need to use those clues to unlock the final answer. The expression is like the treasure, and the equations are the map that leads us there. So, how do we begin? Well, we start by working with the equations one by one and combining them in smart ways.
Let's get started. We have the equations and the expression, and now we also have the understanding of the problem. It is time to put our thinking caps on. Remember that our goal is to solve for the value of the expression, and we are going to do that by taking each of the equations we have to find the values of the variables.
Breaking Down the Equations
Let's tackle each equation to see what we can find. The first equation, a + 3b = 7, gives us a direct relationship between a and b. Since a and b are natural numbers, we can deduce some possibilities. We know that 3b must be less than 7 (because a is positive). This tells us that b can only be 1 or 2, as if b is 3 or greater, then 3b will be 9 or higher, which can not happen. Let's see what happens if b = 1, then a + 3(1) = 7, which gives us a = 4. If we set b = 2, then a + 3(2) = 7, and a = 1. So we have two possible solutions for a and b: (4, 1) and (1, 2).
Moving on to the second equation, 3b + 2c = 9, we can see that since b and c are natural numbers, 3b must be less than 9. We have two possible solutions, let's check both of them. If b = 1, then 3(1) + 2c = 9, so 2c = 6, and c = 3. If b = 2, then 3(2) + 2c = 9, so 2c = 3. Since c has to be a natural number, then the second option doesn't work, which means we can only have b = 1, and c = 3.
The final equation, c + d = 1, is the simplest. Since c and d are natural numbers, the only possibility here is that both of the variables will be 0. But since the question says that they are natural numbers, and natural numbers start from 1, this means that both c and d are not natural numbers, so this equation cannot be satisfied. We made a mistake, let's fix it.
We know that c and d are natural numbers. From the second equation we learned that we could only have b = 1, and c = 3. Let's use it on the third one: 3 + d = 1. Now we can see what the problem is. If we subtract 3 from each side, then we get d = -2. So d is not a natural number. Looking back, we see that the second equation can also not be satisfied because when we found the value of b = 1 and c = 3, we used the first and the second equation separately, and that is not the way to do it.
Let's try a new approach. The first equation is a + 3b = 7. Let's isolate a. Then a = 7 - 3b. Since a is a natural number, 7-3b must be bigger than 0. Now we know the upper value of b, which is 2. The second equation is 3b + 2c = 9. Let's isolate c. c = (9-3b)/2. The third equation is c + d = 1. But we know that c and d are natural numbers. That is only possible if c = 0 and d = 1. But since c has to be a natural number, then we can see that the second equation can't be solved either. Let's try again from the beginning, checking the values for b in both the first and second equations.
The Correct Approach
Let's begin again. From the first equation, we know that a + 3b = 7. From the second equation, 3b + 2c = 9. From the third, c + d = 1. Let's start with the third, since it is the simplest. c and d have to be natural numbers. The only possible way to satisfy this equation is if c = 0 and d = 1. However, since they are natural numbers, c and d cannot be 0. This means that there is no solution to this equation. Since there is no solution, the expression can't be solved either. But let's pretend that c and d were integers. This will show us how to solve the equations if there were a solution. The next step is to find b. We can extract b from both equations. From the first equation, b = (7-a)/3. From the second, b = (9-2c)/3. The third equation gives us c = 1 - d. Substituting c into the second equation, we have 3b + 2(1-d) = 9. 3b + 2 - 2d = 9. 3b = 7 + 2d. This can give us b = (7+2d)/3. So we can substitute that into the first equation, since we know that a = 7-3b, so a = 7 - 3((7+2d)/3). So a = 7 - 7 - 2d = -2d. Since a has to be bigger than 0, then we know that d has to be negative. So we can say that the answer is