Question

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through (4, −7) and perpendicular to the line whose equation is x − 2y – 3 = 0

Accepted Answer

Identify the slope of the given line by rewriting its equation $$x - 2y - 3 = 0 in $$slope-intercept form $$y = mx + b. $$Start by isolating $$y$$: add $$2y to $$both sides and subtract $$3$$ from both sides to get $$x - 3 = 2y$$, then divide both sides by 2 to find $$y = \frac{1}{2}x - \frac{3}{2}$$, so the slope $$m of $$the given line is $$\frac{1}{2}. $$Determine the slope of the line perpendicular to the given line. Recall that perpendicular slopes are negative reciprocals, so if the original slope is $$\frac{1}{2}$$, the perpendicular slope will be $$-2. $$Use the point-slope form of a line equation, which is $$y - y_1$ = m(x - x_1$)$$, where $$(x_1$, y_1$) is $$the point the line passes through and $$m is $$the slope. Substitute the point $$(4, -7)$$ and the perpendicular slope $$-2$$ into the formula to get $$y - (-7) = -2(x - 4). $$Simplify the point-slope form equation to make it clearer: $$y + 7 = -2(x - 4). $$Convert the point-slope form to the general form $$Ax + By + C = 0 by $$distributing and rearranging terms. Start by expanding the right side: $$y + 7 = -2x + 8$$, then bring all terms to one side to get $$2x + y - 1 = 0.$$

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through (4, −7) and perpendicular to the line whose equation is x − 2y – 3 = 0

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Concetti chiave

Point-Slope Form of a Line

Slope of a Line from General Form

Perpendicular Lines and Their Slopes