# What is the point of intersection of 3x + 19y =7 and x + 2y = 1

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To determine the point of intersection of the two lines given by the equations 3x + 19y =7 and x + 2y = 1 we solve the system of equations. The equations given are:

3x + 19y =7… (1)

x + 2y = 1… (2)

Now subtracting 3 times equation 2 from equation 1,

(1)- 3*(2)

=> 3x + 19y- 3x - 6y = 7-3

=> 13y = 4

=> y =4/13

Substituting y = 4/13 in x + 2y = 1

x = 1- 2*4/13

= 5/13

Therefore x= 5/13 and y= 4/13.

**The point of intersection is ( 5/13 , 4/13)**.

To find the point ofintersection of the lines:

3x+19y = 7 and

x+2y = 1.

The cooordinates of any point on a line should satisfy the equation of the line. Therefore the coordintes of point of intersection of the lines should satisfy both the llines. In other words, if we solve equation simultaneously, the solution is the coodintes of the intersection point of the given lines:

We solve by substitution method.

3+19y = 7. Therefore 3x = 7-19y. So , x = (7-19y)/3. We substitute x= (7-19y)/3 in the other equation x+2y = 1.

(7-19y)/3+2y = 1.

Multiply by 3 :

7-19y+6y = 3

7 -13y = 3.

-13y = 3-7 = -4.

y = = -4/-13 = 4/13.

Substitute y = 4/13 in 3x+19y = 7: 3x+19(4/13) = 7 :

3x = 7-19(4/13) = 15/13

x = (15/13)/3 = 5/13

Therefore x = 5/13 and y = 4/13.

Therefore the point of intersection is P whose coordinates are (5/13 , 4/13).

The point of intersection of the two lines forms a part of both the lines. Therefor the values of the coordinates x and y of the point will satisfy both the equations. We can find the value of these coordinates of the point by solving the the two given equations of lines as simultaneous equations.

Given equations are:

3x + 19y = 7 ... (1)

x + 2y = 1 ... (2)

Multiplying equation (2) by 3:

3x + 6y = 3 ... (3)

Subtracting equation (3) from equation (1):

3x - 3x + 19 y - 6y = 7 - 3

==> 13 y = 4

==> y = 4/13

Substituting the above value of y in equation (2):

x + 2*4/13 = 1

==> x = 1 - 8/13 = 5/13

Answer:

Point of intersection is (5/13, 4/13)

(1) y= (7-3x)/19

(2) y= (1-x)/2

Intersection y1=y2 (7-3x)/19=(1-x)/2

2(7-3x)=19(1-x)

14-6x=19-19x 13x=5 x=5/13

(2) y=(1-5/13)/2=4/13

The intersection point is (5/13;4/13)