**Introduction to the equation 4r+3s=75**

In the world of mathematics, equations play a vital role in solving problems and understanding relationships between variables. One such equation that often perplexes students is 4r+3s=75. The equation involves two variables, ‘r’ and ‘s’, and a constant value of 75. In this article, I will guide you through the process of finding the missing variable ‘s’ when the value of ‘r’ is given as 9. By understanding the fundamentals of variables and applying algebraic principles, we can crack this code and unveil the value of ‘s’.

**Understanding variables in equations**

Before we dive into solving the equation, it is essential to grasp the concept of variables. In mathematics, variables are symbols that represent unknown quantities or values that can change. In our equation, ‘r’ and ‘s’ are the variables. The coefficient ‘4’ before ‘r’ signifies that ‘r’ is multiplied by 4, while the coefficient ‘3’ before ‘s’ implies that ‘s’ is multiplied by 3. The equation states that the sum of these two products equals 75. Our goal is to determine the value of ‘s’ when ‘r’ is known to be 9.

**Solving for ‘s’ when given the value of ‘r’**

To find the missing variable ‘s’ in the equation 4r+3s=75, we need to substitute the given value of ‘r’ into the equation. Since we know that ‘r’ is equal to 9, we can replace ‘r’ with 9 in the equation, resulting in 4(9) + 3s = 75.

**Substituting the given value of ‘r’ into the equation**

Now that we have substituted the given value of ‘r’ into the equation, we can simplify it further. Multiplying 4 by 9 gives us 36, so our equation now becomes 36 + 3s = 75.

**Simplifying the equation to solve for ‘s’**

To solve for ‘s’, we need to isolate it on one side of the equation. We can achieve this by subtracting 36 from both sides of the equation. This step leaves us with 3s = 39.

Next, we divide both sides of the equation by 3 to obtain the value of ‘s’. Dividing 39 by 3 yields s = 13.

**Step-by-step calculation to find the value of ‘s’**

To summarize the steps we’ve taken so far:

- Start with the equation 4r + 3s = 75.
- Substitute the given value of ‘r’ into the equation: 4(9) + 3s = 75.
- Simplify the equation: 36 + 3s = 75.
- Isolate ‘s’ by subtracting 36 from both sides: 3s = 39.
- Solve for ‘s’ by dividing both sides by 3: s = 13.

**Explanation of the solution and its significance**

After carefully following the steps outlined above, we have successfully found the value of ‘s’. In the equation 4r + 3s = 75, when ‘r’ is equal to 9, ‘s’ equals 13. This means that when ‘r’ is 9, the equation holds true if ‘s’ is 13. The solution to the equation allows us to understand the relationship between ‘r’ and ‘s’ and how they contribute to the overall value of 75.

**Using algebraic principles to verify the solution**

To ensure the accuracy of our solution, we can employ algebraic principles to verify it. By substituting the values of ‘r’ and ‘s’ back into the original equation, we can confirm that it holds true.

When ‘r’ is 9 and ‘s’ is 13, the equation becomes 4(9) + 3(13) = 75. Simplifying further, we have 36 + 39 = 75, which indeed verifies the correctness of our solution.

**Other examples of equations with missing variables**

Understanding how to solve equations with missing variables is a fundamental skill in mathematics. Let’s explore a couple of other examples to solidify our knowledge:

Example 1: 2x + 5 = 17 To find the value of ‘x’ in this equation, we can start by subtracting 5 from both sides: 2x = 12. Then, divide both sides by 2 to solve for ‘x’: x = 6.

Example 2: 3y – 7 = 16 To solve for ‘y’ in this equation, we add 7 to both sides: 3y = 23. Dividing both sides by 3 gives us the value of ‘y’: y = 7.67.

**Conclusion: Importance of understanding variables in equations**

In conclusion, understanding variables and how they interact in equations is crucial in solving mathematical problems. By following the step-by-step process, we successfully found the missing variable ‘s’ in the equation 4r + 3s = 75 when ‘r’ is known to be 9. This knowledge can be applied to various scenarios where equations with missing variables arise. Remember, practice is key to mastering this skill and unlocking the potential of equations.

*Now that you’ve learned how to find the missing variable ‘s’ in the equation 4r + 3s = 75, why not challenge yourself with more equations? Expand your mathematical horizons and continue exploring the world of variables and equations!*