Write a equation in standard form that satisfies the given conditions?

Question

1) parallel to y=4x+1, and passes through (-3,5)

3) perpendicular to 3x+4y=12 and passes through (7,1)

5) parallel to the x- axis and passes through (4,-1)

Answer 1

NUMBER ONE

The equation is given in slope-intercept form, y = mx + b. m is the slope of the line, and b is the y-intercept (the point where the line crosses the y-axis).

A line that is parallel to another line has the same slope but a different y-intercept.

So, the line parallel to y = 4x+1 will have the same slope, m = 4.

So then, the parallel line has the equation y = 4x + b.

Use the point given in the problem.  Plug (-3, 5) into x and y, and then solve for b.

y = 4x + b

5 = 4(-3) + b

5 = -12 + b

17 = b.  Plug the value of b into the equation.

y = 4x + 17.  Now, rewrite the equation in standard form, Ax + By = C.  Subtract y from both sides:

0 = 4x - y + 17.  Subtract 17 from both sides:

-17 = 4x - y

4x - y = -17

The line parallel to y = 4x + 1 and that passes through (-3,5) has the equation 4x - y = -17

NUMBER THREE

Now, the equation is given in standard form.  Change this to slope-intercept form by solving for y:

3x + 4y = 12

4y = -3x + 12

y = (-3/4)x + 3

The slope of the perpendicular line is the negative reciprocal of the slope of the first line.

The slope of the above equation is -3/4.  The negative reciprocal of -3/4 is 4/3.  So, the slope of the perpendicular line is m = 4/3.

So, the equation of the perpendicular line is y = (4/3)x + b.

Again, plug the point given into the equation and solve for b:

y = (4/3)x + b, point (7, 1)

1 = (4/3)(7) + b

1 = (28/3) + b

-25/3 = b

y = (4/3)x - (25/3).  Rewrite this in standard form:

y = (4/3)x - (25/3)

25/3 = (4/3)x - y.  Multiply both sides of the equation by 3 to get rid of the fractions.

3 * (25/3) = 3 * (4/3)x - 3y

25 = 4x - 3y

The line perpendicular to 3x + 4y = 12 and that passes through (7, 1) has the equation 4x - 3y = 25.

NUMBER FIVE

Since the line is parallel to the x-axis, it will be in the form y = ___.

Since, the y-coordinate of the point is -1, y = -1.

The line parallel to the x-axis and that passes through (4, -1) has the equation y = -1.