3117 views

### How do you find the coordinate of where two lines intersect?

Question:

Line A has a gradient of 4 and passes through point (5,6).

Line B passes through points C (0,5) and D (2,0).

Find the coordinates of the point where the two lines intersection.

Solution:

First of all find the equation of line A:

Using y= mx + c,

Applying the gradient, line A has equation, y = 4x + c

To find c, substitute in the coordinates of point P,

6 = (4x5) + c

6 = 20 + c

c = 6 - 20 = -14

Therefore the equation of line A is y = 4x - 14

Now find the equation of line B:

Using ( y2 - y1 ) / ( x2 - x1 ) = gradient of a line

Substitute in coordinates of points C and D,

( yC - yB ) / ( xC - x) = ( 5 - 0 ) / ( 0 - 2 ) = 5/-2 or -5/2

Using y = mx + c

Applying the gradient found, line B has the equation, y = -5/2 x + c

To find c, substitute in the coordinates of point C,

5 = ( -5/2 x 0 ) + c

c = 5

Therefore the equation of line B is y = -5/2 x + 5

This can be rearranged,

(multiply everything by 2) --> 2y = -5x + 10

(rearrange) ---> 5x + 2y = 10

You can check your answer by using the coordinates of point D,

( 5 x 2 ) + ( 2 x 0 ) = 10 ---> Yes

Finally find the coordinates where the lines intersect:

A   y = 4x - 14

B   5x + 2y = 10

A x2  2y = 8x - 28

Rearrange 8x - 2y = 28

Using simultaneous equations, add A x2 and B, to eliminate y,

5x + 8x + 2y - 2y = 10 + 28

13x = 38

x = 38/13

Substitute in x to A to find y,

y = ( 4 x 38/13 ) - 14

y = 152/13 - 182/13

y = -30/13

Put these coordinates into the equation for line B to check it works,

( 5 x 38/13 ) + (2 x -30/13 ) = 10

190/13 - 60/13 = 130/13 = 10 ----> Yes

The lines cross at coordinate ( 38/13, -30/13 )

3 years ago

Answered by Imogen, an A Level Maths tutor with MyTutor

## Still stuck? Get one-to-one help from a personally interviewed subject specialist

#### 350 SUBJECT SPECIALISTS

£26 /hr

Degree: PGCE Secondary Mathematics (Other) - Leeds University

Subjects offered:Maths, Further Mathematics

Maths
Further Mathematics

“I am currently completing 2 PGCEs in Leeds. I have always had a passion for maths and my objective is to help as many as possible reach their full potential.”

£26 /hr

Degree: Mathematics (Masters) - Sheffield University

Subjects offered:Maths, Further Mathematics + 3 more

Maths
Further Mathematics
.STEP.
.MAT.
-Personal Statements-

“I am a fun, engaging and qualified tutor. I'd love to help you with whatever you need, giving you the support you need to be the best you can be!”

£26 /hr

Degree: Mathematical and Theoretical Physics (Masters) - Oxford, Merton College University

Subjects offered:Maths, Science+ 5 more

Maths
Science
Physics
Further Mathematics
Chemistry
.MAT.
-Personal Statements-

“Mathematics and Theoretical Physics, University of Oxford. I enjoy sharing my experience and enthusiasm in Maths with those who could do with some help”

Currently unavailable: for new students

Degree: Chemistry (Masters) - University College London University

Subjects offered:Maths, Chemistry

Maths
Chemistry

“I am very enthusiastic and patient. I want my students to be able to understand their studies so that they can enjoy them as much as I do.”

### You may also like...

#### Other A Level Maths questions

Express x^2 + 5x + 10 in the form (x+p)^2 +q

Does the equation: x^2+5x-6 have two real roots? If so what are they?

Showing all your working, evaluate ∫ (21x^6 - e^2x- (1/x) +6)dx

How do I solve this inequality: x^2>2x ?

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this.