A line is such that it's segment between the lines 4x+3y-21=0 and 10x+y-59=0 is bisected at point (4,6). So find the equation of the line?

To find the equation of the line that is bisected at the point (4, 6) between the two given lines, we will follow a systematic approach. We need to determine the coordinates of the points where our line intersects the two lines, and then use the midpoint formula to establish the relationship between these points and the point (4, 6).

Step 1: Determine the Intersection Points

We start with the two equations of the lines:

• Line 1: 4x + 3y - 21 = 0
• Line 2: 10x + y - 59 = 0

First, we can rearrange these equations into slope-intercept form (y = mx + b) to find their slopes and y-intercepts, which may help us visualize their relative positions.

Rearranging Line Equations

For Line 1:

3y = -4x + 21

y = -$\frac{4}{3}$x + 7

For Line 2:

y = -10x + 59

Step 2: Finding the Points of Intersection

Next, we need to find the points on these lines that are equidistant from the point (4, 6). Let's denote the points where our line intersects Line 1 and Line 2 as A(x1, y1) and B(x2, y2) respectively.

Substituting the Point into Line Equations

We can substitute the x-coordinate of the midpoint into both line equations to find the corresponding y-coordinates.

Using the fact that the midpoint M of points A and B (x1, y1) and (x2, y2) is given by:

M = $(\frac{x1+x2}{2},\frac{y1+y2}{2})$

Setting M = (4, 6), we have:

• $\frac{x1+x2}{2}=4\Rightarrow x1+x2=8$
• $\frac{y1+y2}{2}=6\Rightarrow y1+y2=12$

Step 3: Express y1 and y2 in terms of x1 and x2

We can express y1 and y2 in terms of x1 and x2 using the line equations:

• For Line 1: y1 = -$\frac{4}{3}$x1 + 7
• For Line 2: y2 = -10x2 + 59

Setting Up the System of Equations

Now substituting these into y1 + y2 = 12:

-$\frac{4}{3}$x1 + 7 - 10x2 + 59 = 12

Combining like terms gives:

-10x2 - $\frac{4}{3}$x1 + 66 = 12

-10x2 - $\frac{4}{3}$x1 = -54

Multiplying through by -1 to simplify:

10x2 + $\frac{4}{3}$x1 = 54

Step 4: Solve the System

We already have x1 + x2 = 8. We can express x2 in terms of x1:

x2 = 8 - x1

Substituting into the previous equation:

10(8 - x1) + $\frac{4}{3}$x1 = 54

80 - 10x1 + $\frac{4}{3}$x1 = 54

-10x1 + $\frac{4}{3}$x1 = -26

Combining terms gives:

$-\frac{30}{3}x1+\frac{4}{3}x1=-26$

$-\frac{26}{3}x1=-26$

x1 = 3

Substituting back, we find:

x2 = 8 - 3 = 5

Step 5: Finding Corresponding y-coordinates

Now, we can find y1 and y2:

y1 = -$\frac{4}{3}$(3) + 7 = -4 + 7 = 3

y2 = -10(5) + 59 = -50 + 59 = 9

Step 6: Finding the Equation of the Line

Now we have the points A(3, 3) and B(5, 9). The slope (m) of the line through these points is given by:

m = $\frac{y2-y1}{x2-x1}=\frac{9-3}{5-3}=\frac{6}{2}=3$

Using the point-slope form of the equation of a line, y - y1 = m(x - x1), we can use point A(3, 3):

y - 3 = 3(x - 3)

y - 3 = 3x - 9

y = 3x - 6

Final Equation

Thus, the equation of the line that is bisected at the point (4, 6) between the two given lines is:

y = 3x - 6