Need some help with this question

Question

The lengths of three sides of a regular hexagon are (12x+9y) cm, (7x+6y+3) cm and (4x+5y+2) cm. Find,
(a) the values of x and y
(b) the perimeter of the figure

Answer 1

Hi Amy.

 

The key term in this question is "regular." That tells you all sides are of equal length.

 

This means that:

 

12x + 9y = 7x + 6y + 3 = 4x + 5y + 2

 

You can rewrite this as two sets equations:

 

Set 1:

7x + 6y + 3 = 12x + 9y

4x + 5y + 2 = 12x + 9y

 

Set 2:

12x + 9y = 7x + 6y + 3

4x + 5y + 2 = 7x + 6y + 3

 

Multiply the bottom row of each set by -1 and add them to the row above

 

Set 1:

 

7x + 6y + 3 = 12x + 9y
-4x - 5y - 2 = -12x - 9y

 

= 3x + y + 1 = 0

 

Set 2:

 

12x + 9y = 7x + 6y + 3
-4x - 5y - 2 = -7x - 6y - 3

 

= 8x + 4y - 2 = 0

 

Take the resulting equations in each set and put them in a new set we'll call

 

Set 3

 

3x + y + 1 = 0

8x + 4y - 2 = 0

 

Now you have 2 equations, 2 unknowns. Just what we want.

 

Multiply row 1 by -4 and add it to row 2. This will eliminate y.

 

-12x - 4y -4 = 0

8x + 4y - 2 = 0

 

-4x - 6 = 0

 

-4x = 6

 

x = -3/2

 

Plug this value of x back into one of your original equations to solve for y.

 

8(-3/2) + 4y - 2 = 0

 

-12 + 4y - 2 = 0

 

4y = 14

 

y = 7/2

 

Plug these values back into your original equations for the side lengths and see if they work out:

12x + 9y = 12(-3/2) + 9(7/2) = -36/2 + 63/2 = 27/2

7x + 6y + 3 = 7(-3/2) + 6(7/2) + 3 = -21/2 + 42/2 + 6/2 = 27/2

4x + 5y + 2 = 4(-3/2) + 5(7/2) + 2 = -12/2 + 35/2 + 4/2 = 27/2

 

Looks like they check out.

 

Your side length is 27/2 cm. Since the perimeter of a hexagon consists of 6 side lengths:

 

P = 6*27/2 = 81cm.

 

Hope this helps.

Answer 2

cool since its a regular hexagon all sides are equal meaning

 

 

12x + 9y =  7x +6y + 3

 

&

 

7x + 6y + 3 = 4x + 5y + 2

 

Now we have two linear equations which are 

 

5x + 3y = 3

 

& 

 

3x + y = -1 

 

Finally solve by either substitution or elimination im going to use elimination

 

 

multiply the second equation by -3

 

-9x - 3y =3

 

and add it to the first one

 

-4x = 6 

 

x = -1.5

 

y = -1 + 4.5 = 3.5

 

Now substitute these values into the length of  a side since all our sides are equal subbing into one is sufficient

 

12(-1.5) + 9(3.5) = -18 + 31.5 = 13.5

 

 

So our perimeter is 13.5 * 6 = 81 cm those are your answers hope it works good luck