Use the given conditions to write an equation for each line in point-slope form and general form. Passing through (5, −9) and perpendicular to the line whose equation is x + 7y - 12= 0

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Identify the slope of the given line. Rewrite the equation x + 7y - 12 = 0 in slope-intercept form (y = mx + b) by isolating y. Start by subtracting x and adding 12 to both sides, then divide through by 7 to solve for y.

Determine the slope of the line perpendicular to the given line. Recall that the slopes of perpendicular lines are negative reciprocals. If the slope of the given line is m, the slope of the perpendicular line will be -1/m.

Use the point-slope form of a line equation, which is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. Substitute the point (5, -9) and the perpendicular slope found in the previous step into this formula.

Simplify the point-slope form equation to write it in general form, Ax + By + C = 0. Expand the equation, combine like terms, and rearrange it so that all terms are on one side of the equation.

Ensure the coefficients in the general form are integers (if necessary, multiply through by a common denominator to eliminate fractions) and simplify further to finalize the equation.

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Point-Slope Form

Point-slope form is a way to express the equation of a line given a point on the line and its slope. The formula is written as y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful for quickly writing the equation of a line when you know a specific point and the slope.

Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1. This means that if one line has a slope of m, the slope of the line perpendicular to it will be -1/m. Understanding this relationship is crucial for finding the slope of the line that is perpendicular to a given line, which is necessary for writing the equation in point-slope form.

General Form of a Line

The general form of a line is expressed as Ax + By + C = 0, where A, B, and C are constants. This form is useful for analyzing the relationship between the coefficients and the line's properties, such as intercepts and parallelism. Converting from point-slope form to general form involves rearranging the equation to fit this standard format.