Question 1 options:
a rectangle with a width of 2x + 3 and a length of 4, has the same area as a triangle with a base of 6 and a height of 3x – 2. solve the following.

x =

area of triangle =

x = 18 units.

Area of triangle = 156 units^2.

Step-by-step explanation:

Area of the rectangle =  4(2x + 3) and the area of the triangle = 1/2*6*(3x - 2).

These are equal, so:

4(2x + 3) = 1/2*6*(3x - 2)

8x + 12 = 9x - 6

12 + 6 = 9x - 8x

x = 18.

Area of triangle =  1/2 * 6 * (3(18) - 2)

= 3 * 52

= 156 units^2.

This means that there exist constant a,b such that if:

                                au+bv=0

                             then a=b=0

{u+v,u-v} is a linearly independent set.

Let us consider there exists constant c,d such that:

                            c(u+v)+d(u-v)=0

To show:   c=d=0

The expression could also be written as:

 cu+cv+du-dv=0

( Since, using the distributive property)

Now on combining the like terms that is the terms with same vectors.

cu+du+cv-dv=0

i.e.

(c+d)u+(c-d)v=0

Since, we are given that u and v are linearly independent vectors this means that:

c+d=0(1)

and c-d=0 i.e c=d(2)

and from equation (1) using equation (2) we have:

2c=0

i.e. c=0

and similarly by equation (2) we have:

         d=0

Hence, we are proved with the result.

We get that the vectors {u+v,u-v} is linearly independent.