write a linear equation to fit each situation or description.
1) the line through (2,4) parallel to the x axis.
2) the line through (2,4) perpendicular to the x axis.
3) the line through (3,1) perpendicular to y=4x-3.
4) the line through (-2,5) parallel to y=-3x.
5) the line through (-1,-4) parallel to 2x-3y=4.
6) the line perpendicular to 4x-y+5=0 at its y intercepts.
i don't understand what it is asking for. can someone please explain and help me?
these are all point-slope problems.
#3: y=4x-3 has slope 4, so your line has slope -1/4, and its equation will be
y-1 = -1/4 (x-3)
are these correct?
1) y=4
2) x=2
4) y=-3x-1
5) y+4=2/3x+1
6) y=4x+5
all are ok, except #5 should be y+4 = 2/3 (x+1)
It makes a difference!
Certainly! I'll explain what each question is asking for and guide you on how to find the linear equations for each situation.
1) The line through (2,4) parallel to the x-axis:
To find a line parallel to the x-axis, we need to remember that all points on the line will have the same y-coordinate. Since the line is passing through the point (2,4), the equation will be y = 4, where y is the y-coordinate.
2) The line through (2,4) perpendicular to the x-axis:
A line perpendicular to the x-axis will have a slope of 0 since it is parallel to the y-axis. Hence, the equation will be x = 2, where x is the x-coordinate.
3) The line through (3,1) perpendicular to y = 4x - 3:
First, let's find the slope of the given line. The slope-intercept form of a line is y = mx + b, where m represents the slope. In the given equation, the coefficient of x is 4, so the slope is 4. The line perpendicular to this will have a negative reciprocal slope, which is -1/4. Using the point-slope form of a linear equation, the equation of the line passing through the point (3,1) with a slope of -1/4 is y - 1 = (-1/4)(x - 3).
4) The line through (-2,5) parallel to y = -3x:
Since the given line has a slope of -3, any line parallel to it will have the same slope of -3. Using the point-slope form, the equation of the line passing through (-2,5) with a slope of -3 is y - 5 = -3(x + 2).
5) The line through (-1,-4) parallel to 2x - 3y = 4:
First, let's rearrange the given equation in slope-intercept form (y = mx + b). Subtracting 2x from both sides gives: -3y = -2x + 4. Dividing throughout by -3 gives y = (2/3)x - 4/3. Since the line is parallel to this, it will have the same slope of 2/3. Using the point-slope form, the equation of the line passing through (-1,-4) with a slope of 2/3 is y + 4 = (2/3)(x + 1).
6) The line perpendicular to 4x - y + 5 = 0 at its y-intercept:
First, let's rearrange the given equation in slope-intercept form (y = mx + b). Adding y and 4x to both sides and rearranging gives y = 4x + 5. The given line has a slope of 4, so any line perpendicular to it will have a negative reciprocal slope of -1/4. We are asked for the line's y-intercept, which occurs when x = 0. Substituting x = 0 into the equation gives y = 5. Thus, the equation of the line perpendicular to 4x - y + 5 = 0 at its y-intercept is y = 5.