Final answer:
1. The equation of the line with a slope of -1 and x-intercept of 6 is y = -x + 6. 2. The equation of the line perpendicular to 3x + 4y - 19 = 0 and passing through (-7,-5) is y = (4/3)x + 7/3. 3. The equation of the line passing through (2,3) and (-1,-2) is y - 3 = (5/3)(x - 2).
Step-by-step explanation:
1. To find the equation of a line with a slope of -1 and an x-intercept of 6, we can use the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Since the line passes through the x-intercept (6,0), we can substitute these values into the equation to solve for b. This gives us the equation y = -x + 6.
2. To find the equation of a line perpendicular to the line 3x + 4y - 19 = 0, we need to find the negative reciprocal of the slope of the given line. The given line can be rewritten in slope-intercept form as y = -(3/4)x + 19/4. The negative reciprocal of -(3/4) is 4/3. So the equation of the perpendicular line passing through the point (-7,-5) is y = (4/3)x + 7/3.
3. To find the equation of a line passing through the points (2,3) and (-1,-2), we can use the point-slope form, y - y1 = m(x - x1), where (x1, y1) is one of the given points and m is the slope. The slope can be calculated as (change in y) / (change in x) = (3 - (-2)) / (2 - (-1)) = 5/3. Substituting the values of x1, y1, and m into the point-slope form, we get the equation y - 3 = (5/3)(x - 2).
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