3x+4y=0
The equation \(3x + 4y = 0\) represents a linear relationship between \(x\) and \(y\). To analyze or plot this equation, we can express it in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
To do this, we'll solve for \(y\):
Start with the original equation: \[ 3x + 4y = 0 \]
Isolate \(y\) by moving \(3x\) to the other side: \[ 4y = -3x \]
Divide both sides by \(4\): \[ y = -\frac{3}{4}x \]
From this form, we can see that the slope (\(m\)) is \(-\frac{3}{4}\), and the y-intercept (\(b\)) is \(0\). This means that the line passes through the origin (0,0) and has a downward slope.
Intercepts
Y-Intercept: Set \(x = 0\): \[ y = -\frac{3}{4}(0) = 0 \quad \text{(point: (0,0))} \]
X-Intercept: Set \(y = 0\): \[ 0 = -\frac{3}{4}x \implies x = 0 \quad \text{(point: (0,0))} \]
Example Points
You can also find additional points by assigning values to \(x\) and calculating \(y\):
If \(x = 4\): \[ y = -\frac{3}{4}(4) = -3 \quad \text{(point: (4,-3))} \]
If \(x = -4\): \[ y = -\frac{3}{4}(-4) = 3 \quad \text{(point: (-4,3))} \]
Summary
The equation \(3x + 4y = 0\) describes a straight line passing through the origin with a slope of \(-\frac{3}{4}\), indicating it falls as \(x\) increases. You can graph these points to visualize the line.
