Questions: Use the given conditions to write an equation for the line in point-slope form and slope-intercept form. Passing through (-3,-3) and (3,7) Type the point-slope form of the equation of the line. (Use integers or simplified fractions for any numbers in the equation.) Type the slope-intercept form of the equation of the line. (Use integers or simplified fractions for any numbers in the equation.)

Solution Steps

To find the equations of the line passing through the points \((-3, -3)\) and \((3, 7)\), we need to follow these steps:

1. Calculate the slope \(m\) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
2. Use the point-slope form of the equation \(y - y_1 = m(x - x_1)\) with one of the given points.
3. Convert the point-slope form to the slope-intercept form \(y = mx + b\).

To find the slope \( m \) of the line passing through the points \((-3, -3)\) and \((3, 7)\), we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points: \[ m = \frac{7 - (-3)}{3 - (-3)} = \frac{10}{6} = \frac{5}{3} \approx 1.6667 \]

Using the point-slope form of the equation \( y - y_1 = m(x - x_1) \) with the point \((-3, -3)\): \[ y - (-3) = \frac{5}{3}(x - (-3)) \] Simplifying: \[ y + 3 = \frac{5}{3}(x + 3) \]

To convert the point-slope form to the slope-intercept form \( y = mx + b \), we expand and simplify: \[ y + 3 = \frac{5}{3}x + \frac{5}{3} \cdot 3 \] \[ y + 3 = \frac{5}{3}x + 5 \] Subtracting 3 from both sides: \[ y = \frac{5}{3}x + 2 \]

Final Answer