what is the equation of straight line passing through intersection of lines 2x+y=3 and x+2y+5=0 , if the intercepts made by the line on positive x axis and positive y axis are in ratio 3:4??
from the intercepts ... the slope is ... -4/3
multiply 1st eqn. by two , then subtract 2nd eqn. ... 3x - 5 = 6
solve for x , then substitute back to find y ... this is the intersection point
use point-slope to write the solution equation
So the answer is
12x+9y=5
Plse reply
To find the equation of the straight line passing through the intersection of two given lines, we need to first find the point of intersection.
Let's solve the given system of equations:
1) 2x + y = 3
2) x + 2y + 5 = 0
To find the point of intersection, we can use the method of substitution or elimination. Let's use the method of substitution.
From equation 1, we have y = 3 - 2x.
Substituting this value into equation 2, we get:
x + 2(3 - 2x) + 5 = 0
x + 6 - 4x + 5 = 0
-3x + 11 = 0
-3x = -11
x = 11/3
Substituting this value back into equation 1, we get:
2(11/3) + y = 3
22/3 + y = 3
y = 3 - 22/3
y = 7/3
Therefore, the point of intersection is (11/3, 7/3).
Now, let's find the intercepts made by this line on the positive x-axis and the positive y-axis, given that their ratio is 3:4.
The intercept on the x-axis is (x, 0), where y = 0.
Substituting y = 0 into the equation of the line passing through the point of intersection:
2x + 0 = 3
2x = 3
x = 3/2
The intercept on the y-axis is (0, y), where x = 0.
Substituting x = 0 into the equation of the line passing through the point of intersection:
0 + y = 3
y = 3
Now, we can calculate the ratio of the intercepts:
Ratio = x-intercept / y-intercept
= (3/2) / 3
= 1/2
Since the given ratio is 3:4, let's scale up the intercepts by multiplying them by 2:
x-intercept = 3/2 * 2 = 3
y-intercept = 3 * 2 = 6
Therefore, the scaled-up intercepts are 3 and 6.
Finally, we can use the point-slope form of the equation of a line to find the equation of the line passing through the point of intersection and having intercepts (3, 6):
y - y1 = m(x - x1), where (x1, y1) is the point of intersection and m is the slope.
Using the point of intersection (11/3, 7/3) and the slope, which is the ratio of the intercepts:
m = (6 - 0) / (3 - 0) = 6/3 = 2
Plugging in the values into the equation, we get:
y - (7/3) = 2(x - 11/3)
Simplifying the equation further, we have:
3y - 7 = 6(x - 11/3)
Expanding the equation, we get:
3y - 7 = 6x - 22
Rearranging the equation to get it in the standard form:
6x - 3y = 15
Therefore, the equation of the straight line passing through the intersection of the lines 2x + y = 3 and x + 2y + 5 = 0, with intercepts on the positive x-axis and y-axis in the ratio 3:4, is 6x - 3y = 15.