Pregunta
Complete the equation of the line through \( (3,-8) \) and \( (6,-4) \). Use exact numbers. \( y=\square \)
Ask by Williams Cummings. in the United States
Jan 25,2025
Solución de tutoría real
Respuesta verificada por el tutor
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The equation of the line is \( y = \frac{4}{3}x - 12 \).
Solución
To find the equation of the line passing through the points \( (3, -8) \) and \( (6, -4) \), follow these steps:
1. **Calculate the slope (\( m \)):**
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - (-8)}{6 - 3} = \frac{4}{3}
\]
2. **Use the slope-intercept form (\( y = mx + b \)) to find the y-intercept (\( b \)):**
Substitute one of the points into the equation. Let's use \( (3, -8) \):
\[
-8 = \frac{4}{3}(3) + b \\
-8 = 4 + b \\
b = -12
\]
3. **Write the final equation:**
\[
y = \frac{4}{3}x - 12
\]
**Final Answer:**
\[
y = \frac{4}{3}x - 12
\]
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Simplifique esta solución Beyond the Answer
To find the equation of the line through the points \( (3,-8) \) and \( (6,-4) \), we first need to determine the slope (m) of the line. The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in our points: \[ m = \frac{-4 - (-8)}{6 - 3} = \frac{4}{3} \] Now, we can use the point-slope form of the line equation, which is \( y - y_1 = m(x - x_1) \). Choosing point \( (3, -8) \): \[ y - (-8) = \frac{4}{3}(x - 3) \] Simplifying it, we get: \[ y + 8 = \frac{4}{3}x - 4 \] \[ y = \frac{4}{3}x - 12 \] So, the completed equation of the line is: \[ y = \frac{4}{3}x - 12 \]
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