how do I find intersections?

Question

where do the lines 6x+2y=26 and 2x+3y=18 intersect?

Answer 1

First, it would be helpful to draw a quick sketch of the lines.  It helps to visualize the problem.

 

To find the intersection point, we need to find the point where x and y are the same value in both equations.

 

The line equations:

 

6x+2y=26  ................... 1

2x+3y=18  ................... 2

 

Can be rearranged to the common line equation form:  y = mx + c

 

y = 13 - 3x ................... 3

y = 6 - 2/3 x ................. 4

 

At the intersection point, y will be equal for both equations.  So, we can set 3 equal to 4 and solve for x.

 

13- 3x = 6 - 2/3 x

13 = 6 + 3x - 2/3x  ....... add 3x to both sides

13 = 6 + 2 1/3x     ........ simplify

7 = 2 1/3 x            ........ subtract 13 from both sides

7 = 7/3 x              ......... multiply both sides by 3/7

3 = x

 

To calculate the y-coordinate substitute x = 3 into 3.

 

y = 13 - 3x

y = 13 - 3(3)

y = 4

 

To check your answer, substitute the values for x and y into the other equation, 4.

 

The point of intersection is (3,4).

 

If you drew a sketch of the problem, you should be able to see that this point of intersection makes sense.

 

Answer 2

 

Hey Micki -- I like "cutting in half" ... 3x +y =13 ... subtract (2x +3y =18) => x- 2y = -5 ... next, add (6x + 2y = 26) => 7x = 21 ... x=3 and y= 13 - 3x = 13 -9 or y=4 ==> (3,4) intersection ... Regards :)

Answer 3

Micky,

 

when determining the point of intersection between two lines, what you must think is that the intersection is the point value that is valid for both equations. So what you want to do is use your two equations to get an equation of just one variable and then solve for the variable.

 

Once you have solved for one variable, substitute it into one of the equations and solve for the other variable.

 

Then to verify you have the right number, take the vales of x and y you got from the first equation, put them into the second equation and verify that it is correct.

 

Using your problem

 

EQN1: 6x+2y=26

EQN2: 2x+3y=18

 

To get the an equation of lets say y you would want multiply both sides of EQN2 by 3 and then subtract EQN2 from EQN1

 

3*(2x+3y) = 3*18

6x+9y = 54  (this is EQN2 multiplied by 3:It will be name EQN2a

 

EQN1-EQN2a

 

    6x + 2y = 26

- (6x + 9y  = 54)

-------------------

     6x + 2y = 26
   - 6x - 9y = -54

-------------------

    0x - 7y = -28  (equation of just y)

 

now solving for y

 

-7y = -28

 

(-7y)/(-7) = (-28)/(-7)

y =4

 

Substitute in the y =4 for y in EQN1

 

6x +(2)*(4) = 26

6x +8 = 26

6x = 18

x = 3

 

So you have determined that your point of intersection is (3,4)

Lets verify that is correct by putting the point into EQN2 and verify that it is true

 

(2)*(3) + (3)*(4) = 18

 

6 + 12 = 18   It is correct so you have the point of intersection

 

I hope this helps