Solving For Y: Expressing Equation As A Function Of X

In this article, we're going to walk through how to rewrite a given equation so that it expresses $y$ as a function of $x$. Basically, we want to isolate $y$ on one side of the equation. Let's dive right into it, guys!

Understanding Functions

Before we get started, let's make sure we understand what it means to express an equation as a function of $x$. A function is a relationship between two variables, where each value of the independent variable ($x$ in this case) corresponds to exactly one value of the dependent variable ($y$). When we write $y$ as a function of $x$, we're essentially writing an equation in the form $y = f(x)$, where $f(x)$ is some expression involving $x$.

Why Express $y$ as a Function of $x$?

Expressing $y$ as a function of $x$ is super useful in many areas of mathematics and its applications. For example:

1. Graphing: When you have $y$ as a function of $x$, it's easy to plot the graph of the function. Just plug in different values of $x$ and calculate the corresponding $y$ values.
2. Calculus: In calculus, you often need to find the derivative of a function. Having $y$ explicitly in terms of $x$ makes differentiation much easier.
3. Modeling: Many real-world phenomena can be modeled using functions. Expressing relationships in functional form helps us understand and analyze these phenomena.

Rewriting the Equation

Alright, let's take the equation $56x + 7y + 21 = 0$ and rewrite it as a function of $x$. Our goal is to isolate $y$ on one side of the equation. Here's how we can do it:

Step 1: Isolate the Term with $y$

First, we want to get the term containing $y$ by itself on one side of the equation. To do this, we subtract $56x$ and $21$ from both sides:

$56x + 7y + 21 - 56x - 21 = 0 - 56x - 21$

This simplifies to:

$7y = -56x - 21$

Step 2: Solve for $y$

Now that we have $7y$ isolated, we need to get $y$ by itself. To do this, we divide both sides of the equation by $7$:

$\frac{7y}{7} = \frac{-56x - 21}{7}$

This simplifies to:

$y = \frac{-56x}{7} - \frac{21}{7}$

Step 3: Simplify the Expression

Finally, we simplify the expression by performing the divisions:

$y = -8x - 3$

So, the equation $56x + 7y + 21 = 0$ rewritten as a function of $x$ is $y = -8x - 3$.

Verification

To make sure we did everything correctly, let's plug the expression for $y$ back into the original equation and see if it holds true.

Original equation: $56x + 7y + 21 = 0$

Substitute $y = -8x - 3$:

$56x + 7(-8x - 3) + 21 = 0$

Distribute the $7$:

$56x - 56x - 21 + 21 = 0$

Simplify:

$0 = 0$

Since the equation holds true, we know that our expression for $y$ as a function of $x$ is correct.

Practical Applications

Understanding how to manipulate equations and express them in different forms has numerous practical applications. Let's explore a few:

Example 1: Linear Equations in Economics

In economics, linear equations are often used to model supply and demand curves. For instance, suppose the demand for a product is given by the equation:

$2p + 5q = 100$

where $p$ is the price and $q$ is the quantity demanded. To analyze how the quantity demanded changes with respect to price, we can express $q$ as a function of $p$:

$5q = 100 - 2p$

$q = \frac{100 - 2p}{5}$

$q = 20 - \frac{2}{5}p$

This tells us that for every unit increase in price, the quantity demanded decreases by $\frac{2}{5}$ units.

Example 2: Physics - Kinematics

In physics, kinematics deals with the motion of objects. Suppose an object's position $s$ at time $t$ is given by the equation:

$s = ut + \frac{1}{2}at^2$

where $u$ is the initial velocity and $a$ is the acceleration. If we want to find the time $t$ at which the object reaches a certain position, we would need to solve for $t$ in terms of $s$, $u$, and $a$. This might involve using the quadratic formula, but the principle of isolating the variable of interest remains the same.

Example 3: Computer Graphics

In computer graphics, equations are used to define shapes and curves. For example, a line can be represented by the equation $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept. If you want to draw this line on a computer screen, you need to calculate the $y$ values for different $x$ values. Expressing $y$ as a function of $x$ makes this calculation straightforward.

Common Mistakes to Avoid

When rewriting equations, it's easy to make mistakes. Here are some common pitfalls to watch out for:

1. Incorrectly Applying Operations: Make sure to perform the same operation on both sides of the equation. For example, if you subtract a term from one side, you must subtract it from the other side as well.
2. Sign Errors: Pay close attention to the signs of the terms. A simple sign error can completely change the equation.
3. Dividing by Zero: Never divide by zero. If you need to divide by a variable, make sure that variable is not equal to zero.
4. Forgetting to Distribute: When multiplying a term by an expression in parentheses, make sure to distribute the term to each part of the expression.

Practice Problems

To solidify your understanding, here are a few practice problems. Try rewriting each equation as a function of $x$:

1. $3x - 2y + 6 = 0$
2. $4x + 8y - 12 = 0$
3. $-2x + 5y + 10 = 0$

Solutions

1. $y = \frac{3}{2}x + 3$
2. $y = -\frac{1}{2}x + \frac{3}{2}$
3. $y = \frac{2}{5}x - 2$

Conclusion

Rewriting equations as functions of $x$ is a fundamental skill in mathematics. It allows us to express relationships between variables in a clear and concise way, making it easier to analyze and solve problems. By following the steps outlined in this article and avoiding common mistakes, you'll be well on your way to mastering this skill. Keep practicing, and you'll become a pro in no time! Whether it's for economics, physics, or computer graphics, this skill is invaluable. So keep at it, guys, and happy solving!