To solve for \( y \) in the equation \( 4x - 3y = 4 \), follow these steps:
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Start with the equation: \[ 4x - 3y = 4 \]
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Isolate the term with \( y \): \[ -3y = 4 - 4x \]
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Divide each term by \(-3\) to solve for \( y \): \[ y = \frac{4 - 4x}{-3} \] \[ y = -\frac{4}{3} + \frac{4}{3}x \]
This can be rearranged to the slope-intercept form \( y = mx + b \): \[ y = \frac{4}{3}x - \frac{4}{3} \]
Graphing the equation
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To graph this line, we can identify the slope and \( y \)-intercept. The slope \( m \) is \( \frac{4}{3} \) and the \( y \)-intercept \( b \) is \(-\frac{4}{3}\).
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Plot the \( y \)-intercept at \( (0, -\frac{4}{3}) \), which is approximately \((0, -1.33)\).
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From the \( y \)-intercept, use the slope to find another point. Since the slope is \( \frac{4}{3} \), for every 3 units you go to the right (positive \( x \)), you go up 4 units (positive \( y \)).
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Starting from \( (0, -\frac{4}{3}) \), if you move 3 units to the right to \( x = 3 \), you move up 4 units to approximately \( y = 1.67 \), giving a point at \( (3, 1.67) \).
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You can also find other points using the x-intercept. Set \( y = 0 \): \[ 0 = \frac{4}{3}x - \frac{4}{3} \] \[ \frac{4}{3}x = \frac{4}{3} \quad \Rightarrow \quad x = 1 \] So the x-intercept is at \( (1, 0) \).
Final Steps
With the points \( (0, -\frac{4}{3}) \), \( (3, 1.67) \), and \( (1, 0) \), you can graph the line. Be sure to draw a straight line through these points.
If you're using a graphing utility, you can input the equation in the form \( y = \frac{4}{3}x - \frac{4}{3} \) and set the window to a standard view, typically \( x: [-10, 10] \) and \( y: [-10, 10] \). This will give you a clear view of the line.
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