Sure! I'd be happy to help you understand how to solve the equation \(3x + 4y = 0\). The goal is to express one variable in terms of the other. Let's go through the steps together.
Step 1: Choose a Variable to Isolate
You can choose to isolate either \(x\) or \(y\). For this example, let's isolate \(y\).
Step 2: Rearrange the Equation
Starting with the equation:
\[ 3x + 4y = 0 \]
We'll move \(3x\) to the other side by subtracting \(3x\) from both sides:
\[ 4y = -3x \]
Step 3: Solve for \(y\)
Now, to get \(y\) by itself, divide both sides by \(4\):
\[ y = \frac{-3x}{4} \]
Step 4: Understand the Relationship
At this point, we have expressed \(y\) in terms of \(x\). This means that for any value of \(x\), you can find the corresponding value of \(y\) by using the formula \(y = \frac{-3x}{4}\).
Graphing the Equation
You can also think about this equation in a geometric sense. The equation \(3x + 4y = 0\) represents a line in the xy-plane. The slope of the line can be found from the rearranged equation \(y = \frac{-3}{4}x\), which tells us that for every increase of \(4\) in \(y\), \(x\) decreases by \(3\).
Step 5: Find Specific Points
If you want to find specific points on the line, you can plug in different values for \(x\) and calculate the corresponding \(y\):
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If \(x = 0\): \[ y = \frac{-3(0)}{4} = 0 \quad \Rightarrow \quad (0, 0) \]
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If \(x = 4\): \[ y = \frac{-3(4)}{4} = -3 \quad \Rightarrow \quad (4, -3) \]
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If \(x = -4\): \[ y = \frac{-3(-4)}{4} = 3 \quad \Rightarrow \quad (-4, 3) \]
Summary
By choosing one variable to isolate, rearranging the equation, and solving for that variable, you can find a relationship between the two variables. In this case, \(y\) is expressed in terms of \(x\), and you now have a way to calculate values based on the equation.
As long as you remember these steps, you'll be able to solve similar equations on your own!