Determine the equation of the line 𝐿 that passes through the point 𝑃(1; −1) and is parallel to the line whose equation is 3𝑥 − 2𝑦 − 6 = 0. Express the line 𝐿 in the form 𝑎𝑥 + 2𝑦 + 5 = 0 and complete the request: • The value of the slope of the line L is: _______ • The value of 𝑎 is: _______

To find the equation of a line parallel to the given line and passing through point P(1, -1), we need to determine the slope of the given line and then use it to construct the equation. The given line has the equation 3x - 2y - 6 = 0. We can rewrite this equation in slope-intercept form (y = mx + b) by solving for y: 3x - 2y - 6 = 0 -2y = -3x + 6 y = (3/2)x - 3 From this equation, we can see that the slope of the given line is 3/2. Any line parallel to this line will have the same slope. Now, let's use the point-slope form of a line to find the equation of the line L that passes through point P(1, -1) and has a slope of 3/2: y - y₁ = m(x - x₁) Substituting the values into the formula: y - (-1) = (3/2)(x - 1) Simplifying: y + 1 = (3/2)(x - 1) y + 1 = (3/2)x - 3/2 To express the line L in the form ax + 2y + 5 = 0, we need to rearrange the equation: 2y + 1 = (3/2)x - 3/2 2y = (3/2)x - 3/2 - 1 2y = (3/2)x - 3/2 - 2/2 2y = (3/2)x - 5/2 4y = 3x - 5 Now, let's multiply both sides by 2 to eliminate the fraction: 4y = 3x - 5 4y - 3x + 5 = 0 Comparing this equation with the desired form ax + 2y + 5 = 0, we can see that a = -3. Therefore, the equation of the line L is -3x + 2y + 5 = 0. The value of the slope of the line L is 3/2, and the value of a is -3.