Let's dive into solving these equations step by step. Equations can sometimes look daunting, but breaking them down into smaller, manageable parts makes the process much easier. We’ll tackle each equation individually to find the solutions. So, let's get started and make sure we understand each step clearly!

Equation 1: iri = 945i

When we're faced with an equation like iri = 945i, our primary goal is to isolate the variable we're trying to solve for. In this case, we want to find the value of 'r'. To do this, we need to get 'r' by itself on one side of the equation. Notice that both sides of the equation have 'i'. Assuming 'i' is not zero, we can divide both sides by 'i' to simplify the equation. This gives us:

r = 945

So, the solution to the first equation is simply that r equals 945. It’s important to note that this solution is valid as long as 'i' is not zero. If 'i' were zero, the original equation would be 0 = 0, which is true regardless of the value of 'r'. However, in most contexts, we assume 'i' is a non-zero variable or constant. This simple algebraic manipulation allows us to quickly find the value of 'r'. Remember, isolating the variable is a key strategy in solving equations. Now that we have found the value of 'r' for the first equation, we can move on to the next one. Understanding these basic principles helps in tackling more complex equations as well. Keep practicing, and you'll become more comfortable with these algebraic manipulations!

Equation 2: 946irm = t

Now, let's tackle the second equation: 946irm = t. Here, we already know from the first equation that r = 945. We can substitute this value into the second equation to simplify it. Replacing 'r' with 945, we get:

946 * i * 945 * m = t

Multiplying 946 by 945, we get:

894970 * i * m = t

So, the equation simplifies to:

894970im = t

This equation tells us that t is equal to 894970 times the product of i and m. Without additional information or another equation relating 'i' and 'm', we cannot find specific values for 'i', 'm', or 't'. However, we have successfully expressed 't' in terms of 'i' and 'm'. This is a common situation in algebra where you might not be able to find a single numerical answer for each variable, but you can express one variable in terms of others. Understanding this relationship is crucial. If, for example, we knew the value of 'i' and 'm', we could easily calculate 't'. Alternatively, if we knew 't' and 'i', we could solve for 'm', and so on. The key is to use the information you have to simplify the equation and express the variables in relation to each other. This step-by-step approach helps clarify the relationships between the variables and provides a clearer understanding of the equation.

Equation 3: 949i = t

Finally, let's look at the third equation: 949i = t. This equation tells us that t is equal to 949 times i. We now have two expressions for 't':

1. t = 894970im (from equation 2)
2. t = 949i (from equation 3)

Since both expressions are equal to 't', we can set them equal to each other:

894970im = 949i

Now, we want to solve for 'm'. Again, assuming 'i' is not zero, we can divide both sides of the equation by 'i':

894970m = 949

Now, divide both sides by 894970 to isolate 'm':

m = 949 / 894970

Simplifying this fraction, we get:

m ≈ 0.0010603

So, m is approximately 0.0010603. Now that we have a value for 'm', we can find the value of 't' using either equation 2 or equation 3. Let's use equation 3:

t = 949i

Since we don't have a specific value for 'i', we can express 't' in terms of 'i'. However, if we had a value for 'i', we could easily calculate 't'. The important thing is that we now have values for 'r' and 'm', and we have expressed 't' in terms of 'i'. This step-by-step approach allowed us to solve the system of equations and find the relationships between the variables. Understanding these techniques is essential for tackling more complex algebraic problems. Remember to always look for ways to simplify the equations and isolate the variables you're trying to solve for. And always double-check your work to ensure accuracy!

Summary of Solutions

To summarize, we have solved the following equations:

1. iri = 945i: r = 945
2. 946irm = t: t = 894970im
3. 949i = t: m ≈ 0.0010603 and t = 949i

So, there you have it! We've successfully navigated through these equations, finding the values of 'r' and 'm', and expressing 't' in terms of 'i'. Remember, guys, the key to solving equations is to break them down, simplify where possible, and isolate the variables. Keep practicing, and you'll become a pro at this in no time! Whether you're dealing with simple algebraic expressions or more complex systems of equations, these fundamental techniques will serve you well. And don't forget, if you ever get stuck, there are plenty of resources available to help you out. Happy solving!