g) With x-intercept -13 and undefined slope, find the equation of the line. h) Find the equation of the line passing through (-5, -1) and parallel to the line 3x + 8y - 4 = 0. i) F... g) With x-intercept -13 and undefined slope, find the equation of the line. h) Find the equation of the line passing through (-5, -1) and parallel to the line 3x + 8y - 4 = 0. i) Find the equation of the line passing through (12, 5) and perpendicular to 7x - 9y = -1.
Understand the Problem
The question is asking for the equations of lines based on given conditions, such as x-intercepts, points through which the lines pass, and relationships with other lines (parallel or perpendicular). We will solve for the equations of the lines based on these geometric properties.
Answer
g) $x = -13$ h) $y = -\frac{3}{8}x - \frac{23}{8}$ i) $y = -\frac{9}{7}x + \frac{143}{7}$
Answer for screen readers
g) The line is given by the equation $x = -13$.
h) The equation of the line is $y = -\frac{3}{8}x - \frac{23}{8}$.
i) The equation of the line is $y = -\frac{9}{7}x + \frac{143}{7}$.
Steps to Solve
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Equation with Undefined Slope The line with an undefined slope is vertical. Its equation is given by the x-coordinate of the x-intercept. Since the x-intercept is -13, the equation of the line is: $$ x = -13 $$
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Finding the Slope of the Given Line To find the equation of the line parallel to $3x + 8y - 4 = 0$, we begin by rewriting it in slope-intercept form. Rearranging gives: $$ 8y = -3x + 4 $$ $$ y = -\frac{3}{8}x + \frac{1}{2} $$ The slope of this line is $m = -\frac{3}{8}$. Since parallel lines have the same slope, we will use this slope for our new line.
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Using the Point-Slope Form Since we know the slope and a point on the line, we can use the point-slope form to find the equation. For the point (-5, -1) and using $m = -\frac{3}{8}$: $$ y - y_1 = m(x - x_1) $$ Substituting gives: $$ y - (-1) = -\frac{3}{8}(x - (-5)) $$ $$ y + 1 = -\frac{3}{8}(x + 5) $$
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Simplifying the Parallel Line Equation Now, simplify to find the equation in slope-intercept form: $$ y + 1 = -\frac{3}{8}x - \frac{15}{8} $$ $$ y = -\frac{3}{8}x - \frac{15}{8} - 1 $$ $$ y = -\frac{3}{8}x - \frac{15 + 8}{8} $$ $$ y = -\frac{3}{8}x - \frac{23}{8} $$
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Finding the Slope of the Given Line for Perpendicularity For the line $7x - 9y = -1$, we convert to slope-intercept form: $$ -9y = -7x - 1 $$ $$ y = \frac{7}{9}x + \frac{1}{9} $$ The slope here is $m = \frac{7}{9}$. The slope of the line perpendicular to this is the negative reciprocal: $$ m_{perpendicular} = -\frac{9}{7} $$
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Using the Point-Slope Form for Perpendicular Line Using the point (12, 5) with $m = -\frac{9}{7}$: $$ y - 5 = -\frac{9}{7}(x - 12) $$ Simplifying gives: $$ y - 5 = -\frac{9}{7}x + \frac{108}{7} $$ $$ y = -\frac{9}{7}x + \frac{108}{7} + 5 $$ Expressing 5 as a fraction: $$ y = -\frac{9}{7}x + \frac{108 + 35}{7} $$ $$ y = -\frac{9}{7}x + \frac{143}{7} $$
g) The line is given by the equation $x = -13$.
h) The equation of the line is $y = -\frac{3}{8}x - \frac{23}{8}$.
i) The equation of the line is $y = -\frac{9}{7}x + \frac{143}{7}$.
More Information
- For part g, a line with an undefined slope is vertical and represented by its x-coordinate.
- In part h, parallel lines share the same slope, which is crucial in determining the new line's equation.
- In part i, the relationship between slopes of perpendicular lines utilizes negative reciprocals.
Tips
- Confusing the slopes of parallel and perpendicular lines is common; remember that parallel lines have the same slope, whereas perpendicular lines have slopes that are negative reciprocals of each other.
- When rewriting lines to slope-intercept form, ensure you perform arithmetic accurately to avoid sign errors.
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