Find the equation of line passing through pont of intersection of the lines 4x+7y-3=0 and 2x-3y+1=0 that has equal intercept on the axes.
To find the equation of the line passing through the point of intersection of the lines 4x + 7y - 3 = 0 and 2x - 3y + 1 = 0 with equal intercept on the axes, follow these steps:
Step 1: Find the point of intersection of the two lines.
To find the point of intersection, equate the two equations and solve for x and y:
4x + 7y - 3 = 0 (Equation 1)
2x - 3y + 1 = 0 (Equation 2)
Solve equations 1 and 2 simultaneously:
Multiply Equation 2 by 2 to eliminate the x term:
4x - 6y + 2 = 0
Now, add the modified Equation 2 to Equation 1:
4x + 7y - 3
+ 4x - 6y + 2
_____________
8x + y - 1 = 0 (Equation 3)
Solve Equation 3 for x:
8x + y - 1 = 0
8x = 1 - y
x = (1 - y)/8
Substitute x in terms of y into Equation 1:
4((1 - y)/8) + 7y - 3 = 0
Simplify:
(1 - y)/2 + 7y - 3 = 0
1 - y + 14y - 6 = 0
13y - 5 = 0
13y = 5
y = 5/13
Substitute y back into Equation 1, to find x:
4x + 7(5/13) - 3 = 0
4x + 35/13 - 39/13 = 0
4x - 4/13 = 0
4x = 4/13
x = 1/13
So, the point of intersection is (1/13, 5/13).
Step 2: Find the slope of the line passing through the point of intersection and the origin (0, 0).
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
slope = (y2 - y1)/(x2 - x1)
Using (1/13, 5/13) as one point and (0, 0) as the other, the slope of the line can be calculated as follows:
slope = (5/13 - 0)/(1/13 - 0)
slope = (5/13)/(1/13)
slope = 5/1
slope = 5
Step 3: Write the equation of the line.
The equation of a line with slope m and passing through a point (x1, y1) is given by the point-slope form:
y - y1 = m(x - x1)
Using the point (1/13, 5/13) as (x1, y1) and the slope 5 as m, the equation of the line that passes through the point of intersection of the given lines and has equal intercepts on the axes can be written as:
y - 5/13 = 5(x - 1/13)
Multiply through by 13 to eliminate fractions:
13y - 5 = 65x - 5
Rearrange the equation:
65x - 13y = 0
Therefore, the equation of the line passing through the point of intersection of the lines 4x + 7y - 3 = 0 and 2x - 3y + 1 = 0, with equal intercepts on the axes, is 65x - 13y = 0.
The lines intersect at (1/13,5/13)
If the line has equal intercepts, then its slope is -1
So, now you have point and a slope, so use the point-slope form for a line and you get
y - 5/13 = -1(x - 1/13)
you can massage that into whatever form you like.