Convert Y-3=(7/9)(x-2/5) To General Form

Getting mathematical equations into their general form can sometimes feel like a puzzle, but it's a really useful skill to have, especially when you're working with lines. The general form of a linear equation is typically written as $Ax + By = C$, where A, B, and C are integers, and A is usually non-negative. Our starting point is the equation y-3=rac{7}{9}{x-\frac{2}{5}}$. This is in point-slope form, which is super helpful for graphing, but we're aiming for that clean, organized $Ax + By = C$ format. Don't worry if fractions and parentheses look a bit intimidating; we'll break it down step-by-step. The key is to systematically eliminate those fractions and rearrange the terms until we achieve the desired structure. We'll be using basic algebraic maneuvers like multiplication, addition, and subtraction. Think of it as tidying up the equation, putting all the $x$ terms on one side, all the $y$ terms on the other, and all the constant numbers on the remaining side. We'll also make sure that all the coefficients are integers, which is a hallmark of the general form. This process not only helps us convert the equation but also solidifies our understanding of how different forms of linear equations relate to each other. It's a foundational concept in algebra that opens doors to solving systems of equations and understanding the geometry of lines in a more profound way. So, let's dive in and conquer this equation, transforming it from its current state into the elegant general form. We'll go through each step with care, ensuring that no detail is overlooked. This journey through algebraic manipulation will equip you with the confidence to tackle similar problems in the future.

Let's begin by tackling those pesky fractions in our equation: $y-3=\frac{7}{9}$x-\frac{2}{5}}$. The first thing we want to do is clear the fractions to make the equation easier to manage. To do this, we need to find a common denominator for all the fractions present. In this case, we have denominators of 9 and 5. The least common multiple of 9 and 5 is 45. So, we're going to multiply both sides of the entire equation by 45. This is a crucial step because it ensures that the equality remains balanced. When we multiply the left side by 45, we get $45(y-3)$. On the right side, we distribute the 45 to both terms inside the parentheses $45 \times \frac{7{9}${x-\frac{2}{5}}$$. Let's simplify this. For the term $45 \times \frac{7}{9}$, we can simplify by dividing 45 by 9, which gives us 5. So, this part becomes 5 imes 7 \times (x-\frac{2}{5}\), which is 35(x-\frac{2}{5}\). Now, our equation looks like 45(y-3) = 35(x-\frac{2}{5}\). We're making great progress! The next step is to distribute the numbers we've multiplied in. On the left side, $45(y-3)$ becomes $45y - 135$. On the right side, we need to distribute the 35 to the terms inside the parentheses x-\frac{2}{5}\). So, we have $35 imes x$ and $35 imes -\frac{2}{5}$. The $35 imes x$ is simply $35x$. For $35 imes -\frac{2}{5}$, we can simplify by dividing 35 by 5, which gives us 7. Then, we multiply 7 by -2, resulting in -14. So, the right side of our equation is now $35x - 14$. Our equation has transformed into $45y - 135 = 35x - 14$. We've successfully eliminated all the fractions, and the equation is much cleaner. This systematic approach of clearing denominators is a fundamental technique in algebra that simplifies complex expressions and makes further manipulation much more straightforward. It's like clearing the clutter so you can see the essential structure of the equation more clearly. Remember, the goal is to isolate variables and constants, and getting rid of fractions is a major step in that direction.

Now that we have eliminated the fractions, our equation is $45y - 135 = 35x - 14$. Our next objective is to rearrange this equation into the general form $Ax + By = C$. This means we want all the terms containing variables ($x$ and $y$) on one side of the equation, and all the constant terms on the other side. Typically, in the general form, the $x$ term is positive. Let's follow that convention. We'll start by moving the $35x$ term from the right side to the left side. To do this, we subtract $35x$ from both sides of the equation: $45y - 135 - 35x = 35x - 14 - 35x$. This simplifies to $-35x + 45y - 135 = -14$. Now, we need to move the constant term, -135, from the left side to the right side. We do this by adding 135 to both sides: $-35x + 45y - 135 + 135 = -14 + 135$. This simplifies to $-35x + 45y = 121$. We are very close to the general form! The final requirement for the standard general form is that the coefficient of $x$ (which is $A$) should be non-negative. Currently, our $A$ is -35. To make it positive, we can multiply the entire equation by -1. This will flip the sign of every term: $(-1) \times (-35x + 45y) = (-1) imes 121$. Distributing the -1, we get $35x - 45y = -121$. And there we have it! The equation is now in the general form $Ax + By = C$, where $A = 35$, $B = -45$, and $C = -121$. All coefficients are integers, and the coefficient of $x$ is positive. This systematic rearrangement is a key skill in understanding linear equations and their various representations. It shows how different forms are just different ways of expressing the same underlying relationship between variables. By following these steps, you can confidently convert any linear equation into its general form, ready for further analysis or problem-solving. This process reinforces the understanding of algebraic manipulation and the properties of equality, making you a more proficient mathematician.

In summary, converting the equation $y-3=\frac{7}{9}${x-\frac{2}{5}}$ into general form involved several key algebraic steps. We began by clearing the fractions, which is often the first hurdle in dealing with equations that contain them. To do this, we found the least common multiple of the denominators (9 and 5), which is 45, and multiplied every term in the equation by this value. This step transformed the equation into 45(y-3) = 35(x-\frac{2}{5}\). Following this, we distributed the multipliers on both sides to get $45y - 135 = 35x - 14$. The next crucial phase was rearranging the equation to fit the general form $Ax + By = C$. We achieved this by moving all the variable terms to one side and the constant terms to the other. Specifically, we subtracted $35x$ from both sides and added 135 to both sides, resulting in $-35x + 45y = 121$. The final refinement was to ensure the coefficient of the $x$ term ($A$) was positive, as is conventional for the general form. We accomplished this by multiplying the entire equation by -1, which yielded the final equation in general form: $35x - 45y = -121$. Here, $A=35$, $B=-45$, and $C=-121$. These integer coefficients and the positive $A$ value confirm we have successfully reached the standard general form. This process highlights the power of consistent algebraic manipulation and the importance of understanding the conventions associated with different mathematical forms. It's a foundational skill that underpins much of higher-level mathematics. If you'd like to explore more about linear equations and their forms, a great resource is Khan Academy's section on linear equations, which offers detailed explanations and practice exercises.