The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.

1. Equation of a line through a
point and intersection of
another two lines

2. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).

3. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).
2x  y 1  0
3x  5 y  9  0

4. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).
2x  y 1  0 10 x  5 y  5

3x  5 y  9  0 3x  5 y  9

5. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).
2x  y 1  0 10 x  5 y  5 ()

3x  5 y  9  0 3x  5 y  9

6. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).
2x  y 1  0 10 x  5 y  5 ()

3x  5 y  9  0 3x  5 y  9
7x =  14

7. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).
2x  y 1  0 10 x  5 y  5 ()

3x  5 y  9  0 3x  5 y  9
7x =  14
x  2

8. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).
2x  y 1  0 10 x  5 y  5 ()

3x  5 y  9  0 3x  5 y  9
7x =  14
x  2  2  2   y  1  0
y3

9. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).
2x  y 1  0 10 x  5 y  5 ()

3x  5 y  9  0 3x  5 y  9
7x =  14
x  2  2  2   y  1  0
y3
 the lines intersect at  2,3

10. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).
2x  y 1  0 10 x  5 y  5 ()

3x  5 y  9  0 3x  5 y  9
7x =  14
x  2  2  2   y  1  0
y3
3 2  the lines intersect at  2,3
m
2  1
1

3

11. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).
2x  y 1  0 10 x  5 y  5 ()

3x  5 y  9  0 3x  5 y  9
7x =  14
x  2  2  2   y  1  0
y3
3 2 1
y  2    x  1  the lines intersect at  2,3
m
2  1 3
1

3

12. Equation of a line through a
point and intersection of
another two lines
e.g. Find the equation of the line that passes through the intersection of
2x + y + 1= 0 and 3x + 5y – 9 = 0 and the point (1,2).
2x  y 1  0 10 x  5 y  5 ()

3x  5 y  9  0 3x  5 y  9
7x =  14
x  2  2  2   y  1  0
y3
3 2 1
y  2    x  1  the lines intersect at  2,3
m
2  1 3
1 3y  6  x 1

3 x  3y  7  0

13. Alternatively

14. Alternatively
a1 x  b1 y  c1  k  a2 x  b2 y  c2   0

15. Alternatively
a1 x  b1 y  c1  k  a2 x  b2 y  c2   0
2 x  y  1  k  3x  5 y  9   0

16. Alternatively
a1 x  b1 y  c1  k  a2 x  b2 y  c2   0
2 x  y  1  k  3x  5 y  9   0
1, 2  : 2 1   2   1  k  3 1  5  2   9   0

17. Alternatively
a1 x  b1 y  c1  k  a2 x  b2 y  c2   0
2 x  y  1  k  3x  5 y  9   0
1, 2  : 2 1   2   1  k  3 1  5  2   9   0
5  4k  0

18. Alternatively
a1 x  b1 y  c1  k  a2 x  b2 y  c2   0
2 x  y  1  k  3x  5 y  9   0
1, 2  : 2 1   2   1  k  3 1  5  2   9   0
5  4k  0
4k  5
5
k 
4

19. Alternatively
a1 x  b1 y  c1  k  a2 x  b2 y  c2   0
2 x  y  1  k  3x  5 y  9   0
1, 2  : 2 1   2   1  k  3 1  5  2   9   0
5  4k  0
4k  5
5
k 
4
5
2x  y 1  3x  5 y  9   0
4

20. Alternatively
a1 x  b1 y  c1  k  a2 x  b2 y  c2   0
2 x  y  1  k  3x  5 y  9   0
1, 2  : 2 1   2   1  k  3 1  5  2   9   0
5  4k  0
4k  5
5
k 
4
5
2x  y 1  3x  5 y  9   0
4
8 x  4 y  4  15 x  25 y  45  0

21. Alternatively
a1 x  b1 y  c1  k  a2 x  b2 y  c2   0
2 x  y  1  k  3x  5 y  9   0
1, 2  : 2 1   2   1  k  3 1  5  2   9   0
5  4k  0
4k  5
5
k 
4
5
2x  y 1  3x  5 y  9   0
4
8 x  4 y  4  15 x  25 y  45  0
7 x  21 y  49  0

22. Alternatively
a1 x  b1 y  c1  k  a2 x  b2 y  c2   0
2 x  y  1  k  3x  5 y  9   0
1, 2  : 2 1   2   1  k  3 1  5  2   9   0
5  4k  0
4k  5
5
k 
4
5
2x  y 1  3x  5 y  9   0
4
8 x  4 y  4  15 x  25 y  45  0
7 x  21 y  49  0
x  3y  7  0

23. Alternatively
a1 x  b1 y  c1  k  a2 x  b2 y  c2   0
2 x  y  1  k  3x  5 y  9   0
1, 2  : 2 1   2   1  k  3 1  5  2   9   0
5  4k  0
4k  5
5
k 
4
5
2x  y 1  3x  5 y  9   0
4
8 x  4 y  4  15 x  25 y  45  0 Exercise 5F; 2b, 3b, 6b(i),
7ab (i, iii), 9, 10, 13*
7 x  21 y  49  0
x  3y  7  0 Exercise 5G; 2 to 14 evens, 15*