Final answer:
To find the slope-intercept equation for a line parallel to 4x + 5y = 9 and passing through (4, -2), use the same slope and the point-slope form of a linear equation. The equation is y = (-4/5)x + 6/5. To find the slope-intercept equation for a line perpendicular to 4x + 5y = 9 and passing through (4, -2), use the negative reciprocal of the slope and the point-slope form. The equation is y = (5/4)x - 7.
Step-by-step explanation:
To find the slope-intercept equation for a line parallel to the line 4x + 5y = 9 and passing through the point (4, -2), we need to determine the slope of the given line. The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. Since the line we want is parallel to 4x + 5y = 9, it will have the same slope. To find the slope of this line, we can rearrange the given equation into the slope-intercept form, y = mx + b.
4x + 5y = 9
5y = -4x + 9
y = (-4/5)x + 9/5
So, the slope of the given line is -4/5. Now, we can use the point-slope form of a linear equation, (y - y1) = m(x - x1), to find the equation of the line passing through the point (4, -2).
(y - (-2)) = (-4/5)(x - 4)
y + 2 = (-4/5)x + 16/5
y = (-4/5)x + 16/5 - 2
y = (-4/5)x + 6/5
Therefore, the slope-intercept equation for the line parallel to 4x + 5y = 9 and passing through the point (4, -2) is y = (-4/5)x + 6/5.
To find the slope-intercept equation for a line perpendicular to the line 4x + 5y = 9 and passing through the point (4, -2), we need to determine the negative reciprocal of the slope of the given line. The negative reciprocal of -4/5 is 5/4. We can use the point-slope form of a linear equation again to find the equation of the line passing through (4, -2), but this time using the slope of 5/4.
(y - (-2)) = (5/4)(x - 4)
y + 2 = (5/4)x - 5
y = (5/4)x - 5 - 2
y = (5/4)x - 7
Therefore, the slope-intercept equation for the line perpendicular to 4x + 5y = 9 and passing through the point (4, -2) is y = (5/4)x - 7.