Solving The Linear Equation 9/4x + 1 = 11/2

When you encounter a linear equation like $\frac{9}{4} x+1=\frac{11}{2}$, the primary goal is to isolate the variable, which in this case is 'x'. This means getting 'x' all by itself on one side of the equation. To do this, we'll use a series of inverse operations, much like a detective solving a mystery, by undoing each operation that's being applied to 'x'. Remember, the golden rule of algebra is to perform the same operation on both sides of the equation to maintain its balance. Think of it as a seesaw; whatever you do to one side, you must do to the other to keep it level. We'll start by tackling the constant term, the '+1', and then move on to the coefficient that's multiplying 'x', which is $\frac{9}{4}$. By systematically applying these inverse operations, we can unravel the value of 'x' and find the solution to this mathematical puzzle. This process is fundamental in algebra and forms the basis for solving more complex equations.

Step 1: Eliminate the Constant Term

Our first move in solving the equation $\frac{9}{4} x+1=\frac{11}{2}$ is to remove the constant term (+1) from the left side where the 'x' is located. To undo the addition of 1, we perform the inverse operation, which is subtraction. So, we will subtract 1 from both sides of the equation. This is a crucial step because it helps us to start isolating the term containing 'x'. When we subtract 1 from the left side, the '+1' and '-1' cancel each other out, leaving us with just $\frac{9}{4} x$. On the right side, we need to perform the subtraction: $\frac{11}{2} - 1$. To subtract these fractions, they must have a common denominator. The denominator of 1 is implicitly 1, so we can rewrite 1 as $\frac{2}{2}$. Now, the right side becomes $\frac{11}{2} - \frac{2}{2}$. Subtracting the numerators while keeping the common denominator, we get $\frac{11-2}{2}$, which simplifies to $\frac{9}{2}$. So, after this step, our equation transforms into $\frac{9}{4} x = \frac{9}{2}$. This simplified form brings us closer to finding the value of 'x', as the 'x' term is now free from any additive constants.

Step 2: Isolate the Variable 'x'

Now that we have the equation in the form $\frac{9}{4} x = \frac{9}{2}$, our next objective is to isolate 'x' completely. Currently, 'x' is being multiplied by the fraction $\frac{9}{4}$. To undo this multiplication, we need to perform the inverse operation, which is division. However, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{9}{4}$ is $\frac{4}{9}$. Therefore, we will multiply both sides of the equation by $\frac{4}{9}$. This is where the magic of algebraic manipulation happens. On the left side, when we multiply $\frac{9}{4} x$ by $\frac{4}{9}$, the $\frac{9}{4}$ and $\frac{4}{9}$ cancel each other out (since $\frac{9}{4} \times \frac{4}{9} = \frac{36}{36} = 1$), leaving us with just 'x'. On the right side, we perform the multiplication: $\frac{9}{2} \times \frac{4}{9}$. We can simplify this multiplication by canceling out common factors. The '9' in the numerator of the first fraction and the '9' in the denominator of the second fraction cancel each other out. Similarly, the '2' in the denominator of the first fraction and the '4' in the numerator of the second fraction can be simplified, where 4 divided by 2 is 2. So, the multiplication becomes $1 \times \frac{2}{1}$, which equals 2. Alternatively, multiplying the numerators gives $9 \times 4 = 36$, and multiplying the denominators gives $2 \times 9 = 18$. Then, $\frac{36}{18}$ simplifies to 2. Thus, after performing this multiplication, we find that $x = 2$. This is our final solution.

Step 3: Verification of the Solution

To ensure that our calculated value of $x=2$ is indeed the correct solution for the equation $\frac{9}{4} x+1=\frac{11}{2}$, we must perform a verification step. This involves substituting the value of 'x' back into the original equation and checking if both sides are equal. If they are, then our solution is correct. Let's substitute $x=2$ into the original equation: $\frac{9}{4} (2) + 1$. First, we multiply $\frac{9}{4}$ by 2. This can be written as $\frac{9}{4} \times \frac{2}{1}$. Multiplying the numerators gives $9 \times 2 = 18$, and multiplying the denominators gives $4 \times 1 = 4$. So, we have $\frac{18}{4}$. This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2. This gives us $\frac{9}{2}$. Now, we add the constant term: $\frac{9}{2} + 1$. To add these, we need a common denominator, so we rewrite 1 as $\frac{2}{2}$. The expression becomes $\frac{9}{2} + \frac{2}{2}$. Adding the numerators, we get $\frac{9+2}{2}$, which equals $\frac{11}{2}$. Now, we compare this result to the right side of the original equation, which is also $\frac{11}{2}$. Since $\frac{11}{2} = \frac{11}{2}$, our solution $x=2$ is confirmed to be correct. This verification process is a vital part of problem-solving in mathematics, as it builds confidence in our answers and helps catch any potential errors made during the calculation.

Conclusion

In conclusion, solving the linear equation $\frac{9}{4} x+1=\frac{11}{2}$ involves a systematic approach of isolating the variable 'x' using inverse operations. We began by eliminating the constant term '+1' through subtraction, transforming the equation into $\frac{9}{4} x = \frac{9}{2}$. Subsequently, we isolated 'x' by multiplying both sides by the reciprocal of $\frac{9}{4}$, which is $\frac{4}{9}$. This led us to the solution $x=2$. The crucial step of verification confirmed that our solution is accurate by substituting $x=2$ back into the original equation, yielding $\frac{11}{2}$ on both sides. Mastering these techniques is fundamental for anyone studying algebra, as they provide the tools to tackle a wide range of mathematical problems. If you're looking for more resources on solving linear equations or exploring other mathematical concepts, you can find valuable information on Khan Academy or Math is Fun.