Write an equation for the line in point/slope form and slope/intercept form that has the given condition.
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4. Slope = 3/2 and passes through the origin.
5. x-intercept = 4 and y-intercept = -3
6. Passes through (3, 2) and is parallel to 2x-y=4
7. Passes through (-1, -1) and is perpendicular to y= (5)/(2)x+3
as you know, slope is ∆y/∆x
so, for any point (x,y) on the line containing (h,k) with slope m,
m = ∆y/∆x = (y-k)/(x-h)
or, as some like it,
y = m(x-h) + k
So, for #6,
2x-y = 4
y = 2x-4, so
slope is 2, so
(y-2) = 2(x-3)
y = 2x - 4
Do the others the same way. Com eon back if you get stuck. Be sure to show how you got there.
Sure! I can help you with these equations.
4. Slope = 3/2 and passes through the origin.
To write the equation in point/slope form, we can use the formula: y - y1 = m(x - x1), where (x1, y1) represents the coordinates of a point on the line and m represents the slope.
Since the line passes through the origin (0,0), we can substitute x1 = 0 and y1 = 0 into the equation. Also, the slope m is given as 3/2.
Therefore, the equation in point/slope form is: y - 0 = (3/2)(x - 0), which simplifies to y = (3/2)x.
To write the equation in slope/intercept form (y = mx + b), we need to find the y-intercept (b) of the line. Since the line passes through the origin (0,0), the y-intercept is 0.
Therefore, the equation in slope/intercept form is: y = (3/2)x + 0, which simplifies to y = (3/2)x.
5. x-intercept = 4 and y-intercept = -3
The x-intercept represents the point where the line intersects the x-axis, and the y-intercept represents the point where the line intersects the y-axis.
Given that the x-intercept is 4, we have the point (4,0) on the line. Given that the y-intercept is -3, we have the point (0,-3) on the line.
To write the equation, we can use the point/slope form of the equation. We can calculate the slope (m) using the given points: m = (y2 - y1) / (x2 - x1) = (-3 - 0) / (0 - 4) = -3 / -4 = 3/4.
Using the formula y - y1 = m(x - x1) and substituting (x1, y1) = (4,0), we get y - 0 = (3/4)(x - 4), which simplifies to y = (3/4)x - 3.
6. Passes through (3, 2) and is parallel to 2x - y = 4
To write the equation of a line parallel to a given line, we need to have the same slope.
The given line 2x - y = 4 can be rewritten in slope/intercept form as y = 2x - 4. So its slope is 2.
Since the line we are looking for is parallel, it will also have a slope of 2.
Using the point/slope form of the equation, we can substitute the given point (x1, y1) = (3, 2) and slope m = 2 into the formula y - y1 = m(x - x1).
Therefore, the equation in point/slope form is: y - 2 = 2(x - 3), which simplifies to y - 2 = 2x - 6, and further simplifies to y = 2x - 4.
7. Passes through (-1, -1) and is perpendicular to y = (5/2)x + 3
To write the equation of a line perpendicular to a given line, we need to find the negative reciprocal of the slope of the given line.
The given line y = (5/2)x + 3 has a slope of 5/2. The negative reciprocal of 5/2 is -2/5.
Using the point/slope form of the equation, we can substitute the given point (x1, y1) = (-1, -1) and slope m = -2/5 into the formula y - y1 = m(x - x1).
Therefore, the equation in point/slope form is: y - (-1) = (-2/5)(x - (-1)), which simplifies to y + 1 = (-2/5)(x + 1), and further simplifies to y = (-2/5)x - 2/5 - 1, and finally simplifies to y = (-2/5)x - 7/5.