the distance between two lakes is 300 miles. on a map, the measured distance between the lakes is three-fourths of costs $24.00
1 answer:
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For the red line, we have:
gradient (
) = 1/3
y-intercept = 3
Equation of straight line is given by
, where is the gradient and 
is where the line intercept the y-axis
So we have,
The shaded part is above the line, hence the first inequality is
The black line, we have
gradient,
= 3
y-intercept = -2
, rearranging gives
The shaded region is to the right of the line, hence the inequality is
Note: The first option should be the correct answer but the inequality sign for <span>y > 1/3x + 3 should be </span>
gradient (
y-intercept = 3
Equation of straight line is given by
So we have,
The shaded part is above the line, hence the first inequality is
The black line, we have
gradient,
y-intercept = -2
The shaded region is to the right of the line, hence the inequality is
Note: The first option should be the correct answer but the inequality sign for <span>y > 1/3x + 3 should be </span>
The vectors in form a basis of
if they are mutually linearly independent and span
.
To check for independence, we can compute the Wronskian determinant:
The determinant is non-zero, so the vectors are indeed independent.
To check if they span , you need to show that any vector in
can be expressed as a linear combination of the vectors in
. We can write an arbitrary vector in
as
Then we need to show that there is always some choice of scalars such that
This is equivalent to solving
or the system (in matrix form)
This has a solution if the coefficient matrix on the left is invertible. It is, because
(that is, the coefficient matrix is not singular, so an inverse exists)
Compute the inverse any way you like; you should get
Then
A solution exists for any choice of , so the vectors in
indeed span
.
The vectors in are independent and span
, so
forms a basis of
.