Solving Linear Equations: Find (q, R)!

Hey guys! Let's dive into the world of linear equations and figure out how to solve a system of them. We've got two equations here, and our mission is to find the values of q and r that make both equations true at the same time. Think of it like a treasure hunt where we need to find the specific values that satisfy both clues simultaneously. We'll break down the problem step-by-step, making sure it's super clear and easy to follow. So grab your pencils and let's get started! We are going to find the solution $(q, r)$ to this system of linear equations: $12q + 3r = 15$ and $-4q - 4r = -44$.

Understanding the Problem

Alright, before we jump into the nitty-gritty, let's make sure we're all on the same page. What is a system of linear equations, anyway? Well, it's simply a set of two or more linear equations that we're trying to solve together. Each equation represents a straight line when graphed, and the solution to the system is the point (or points) where all the lines intersect. In our case, we've got two lines, and we're looking for the single point (the values of q and r) where they cross each other. Solving a system of linear equations is a fundamental concept in mathematics, and it pops up everywhere, from basic algebra to advanced physics. So, understanding how to solve these problems is a super valuable skill. We'll be using two main methods to find our solution: substitution and elimination. Both methods aim to isolate one of the variables (either q or r) and then solve for the other. Choosing the right method can sometimes make the problem a lot easier, so we'll discuss the best approach for our specific set of equations.

The Given Equations

Let's take a closer look at the equations we're working with: $12q + 3r = 15$ and $-4q - 4r = -44$. The first equation, $12q + 3r = 15$, tells us a relationship between q and r. For example, if we knew the value of q, we could plug it in and solve for r. Similarly, the second equation, $-4q - 4r = -44$, provides another relationship. The goal is to find values for q and r that work for both equations simultaneously. Both equations are linear, meaning that the highest power of the variables is 1. This is why their graphs are straight lines. This linearity allows us to use specific methods like substitution or elimination to find a definite solution. We can manipulate these equations to eliminate one variable and solve for the other. Then, by substituting the value we find back into one of the original equations, we can solve for the remaining variable. This systematic approach is the key to cracking the code and finding the solution $(q, r)$. So, what approach should we choose, and what are the steps?

Method 1: Elimination

Alright, let's get our hands dirty with the elimination method! This is a cool technique where we manipulate the equations so that either the q or r terms cancel out when we add the equations together. This leaves us with a single variable to solve for, making things much easier. In this case, we'll aim to eliminate q first. The equations are: $12q + 3r = 15$ and $-4q - 4r = -44$. Notice that the q coefficients are 12 and -4, respectively. To make these cancel out when we add the equations, we need to make the q coefficients opposites. We can achieve this by multiplying the second equation by 3. This changes the equation $-4q - 4r = -44$ to $-12q - 12r = -132$. Now we can add the modified second equation to the first equation.

Step-by-Step Elimination

Here’s a step-by-step breakdown of the elimination method: First, we multiply the second equation by 3: $3(-4q - 4r) = 3(-44)$, which simplifies to $-12q - 12r = -132$. Next, we write down our two equations: $12q + 3r = 15$ and $-12q - 12r = -132$. Now, add the two equations together: $(12q - 12q) + (3r - 12r) = (15 - 132)$. This simplifies to $0q - 9r = -117$, or just $-9r = -117$. Finally, divide both sides by -9 to solve for r: $r = -117 / -9$, so $r = 13$. Awesome! We've found the value of r. Now, we need to find the value of q. We can substitute this value back into one of the original equations to find q. Let's try plugging r = 13 into the first equation: $12q + 3(13) = 15$. This gives us $12q + 39 = 15$. Subtract 39 from both sides: $12q = 15 - 39$, so $12q = -24$. Finally, divide by 12: $q = -24 / 12$, therefore $q = -2$. Thus, we have the solution $(q, r) = (-2, 13)$.

Verification

It's always a good idea to double-check your answer to make sure you didn't make any mistakes. Let's substitute $q = -2$ and $r = 13$ back into our original equations to see if they hold true. For the first equation, $12q + 3r = 15$, we get $12(-2) + 3(13) = -24 + 39 = 15$. Great, it works! Now let's check the second equation, $-4q - 4r = -44$. Plugging in our values, we get $-4(-2) - 4(13) = 8 - 52 = -44$. Awesome! Both equations are satisfied, so we know we have the correct solution. It's always satisfying to get to the end of a problem and see your solution verified. And guys, we did it! We successfully used the elimination method to find the solution to the system of equations.

Method 2: Substitution

Now, let's try solving this problem using the substitution method! This method involves solving one of the equations for one variable in terms of the other, and then plugging that expression into the other equation. It's like replacing one variable with its equivalent. Let's start with our first equation, $12q + 3r = 15$. We can isolate q by subtracting 3r from both sides: $12q = 15 - 3r$. Then, divide both sides by 12: $q = (15 - 3r) / 12$. We now have an expression for q in terms of r. Next, we'll substitute this expression for q into the second equation, $-4q - 4r = -44$. The goal is to get an equation with only one variable, which we can then solve. This technique is especially useful when one of the equations is already solved for a variable, which simplifies the substitution process.

Step-by-Step Substitution

Let’s walk through the substitution method step-by-step. First, we have $q = (15 - 3r) / 12$. Now, substitute this expression into the second equation: $-4[(15 - 3r) / 12] - 4r = -44$. Simplify the expression: $-4(15 - 3r) / 12 - 4r = -44$. Which simplifies to $-(15 - 3r) / 3 - 4r = -44$. Multiply both sides by 3 to get rid of the fraction: $-(15 - 3r) - 12r = -132$. Expand: $-15 + 3r - 12r = -132$. Combine like terms: $-15 - 9r = -132$. Add 15 to both sides: $-9r = -117$. Finally, divide both sides by -9 to find r: $r = -117 / -9$, which gives us $r = 13$. We've found the value for r! Now, to find q, we can substitute r = 13 back into our expression for q: $q = (15 - 3(13)) / 12$, which simplifies to $q = (15 - 39) / 12$, and further to $q = -24 / 12$, so $q = -2$. So again, we found the solution $(q, r) = (-2, 13)$.

Verification (Again!)

We always double-check our work! Let's substitute $q = -2$ and $r = 13$ back into the original equations to verify that our solution is correct. Using the first equation: $12q + 3r = 15$, we have $12(-2) + 3(13) = -24 + 39 = 15$, which checks out. Using the second equation: $-4q - 4r = -44$, we have $-4(-2) - 4(13) = 8 - 52 = -44$, which also works! Both equations are satisfied. The results are consistent with the results of the elimination method. Great job, guys! Now you've seen two different methods to solve the same problem.

Conclusion: The Solution

So, what's the solution to this system of linear equations? After using both the elimination and substitution methods, we found that the values of q and r that satisfy both equations are q = -2 and r = 13. We can write the solution as an ordered pair: $(-2, 13)$. This means that when q is -2 and r is 13, both equations in the system are true. We've gone over two common and effective methods for solving systems of linear equations. Both methods are valuable and each might be preferable depending on the specific equations given. The most important thing is to understand the underlying principles and practice, practice, practice! With enough practice, you'll become a pro at solving these types of problems. Remember, the key is to stay organized, double-check your work, and always verify your answer. Congratulations, you've conquered another math problem. Keep up the amazing work, and keep exploring the fascinating world of mathematics!