Calla bought 8 writing utensils for a total of $30. Pens cost $4 each and pencils cost $3 each. How many pens did she buy?
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We have the following equations:
So we are asked to write a system of equations or inequalities for each region and each point.
Part a)
Region Example A
Region B.
Let's take a point that is in this region, that is:
So let's find out the signs of each inequality by substituting this point in them:
So the inequalities are:
Region C.
A point in this region is:
So let's find out the signs of each inequality by substituting this point in them:
So the inequalities are:
Region D.
A point in this region is:
So let's find out the signs of each inequality by substituting this point in them:
So the inequalities are:
Point P:
This point is the intersection of the two lines. So let's solve the system of equations:
Accordingly, the point is:
Point q:
This point is the
of the line:
So let:
Then
Therefore, the point is:
Part b)
The coordinate of a point within a region must satisfy the corresponding system of inequalities. For each region we have taken a point to build up our inequalities. Now we will take other points and prove that these are the correct regions.
Region Example A
The origin is part of this region, therefore let's take the point:
Substituting in the inequalities:
It is true.
Region B.
Let's take a point that is in this region, that is:
Substituting in the inequalities:
It is true
Region C.
Let's take a point that is in this region, that is:
Substituting in the inequalities:
It is true
Region D.
Let's take a point that is in this region, that is:
Substituting in the inequalities:
It is true
So we are asked to write a system of equations or inequalities for each region and each point.
Part a)
Region Example A
Region B.
Let's take a point that is in this region, that is:
So let's find out the signs of each inequality by substituting this point in them:
So the inequalities are:
Region C.
A point in this region is:
So let's find out the signs of each inequality by substituting this point in them:
So the inequalities are:
Region D.
A point in this region is:
So let's find out the signs of each inequality by substituting this point in them:
So the inequalities are:
Point P:
This point is the intersection of the two lines. So let's solve the system of equations:
Accordingly, the point is:
Point q:
This point is the
So let:
Then
Therefore, the point is:
Part b)
The coordinate of a point within a region must satisfy the corresponding system of inequalities. For each region we have taken a point to build up our inequalities. Now we will take other points and prove that these are the correct regions.
Region Example A
The origin is part of this region, therefore let's take the point:
Substituting in the inequalities:
It is true.
Region B.
Let's take a point that is in this region, that is:
Substituting in the inequalities:
It is true
Region C.
Let's take a point that is in this region, that is:
Substituting in the inequalities:
It is true
Region D.
Let's take a point that is in this region, that is:
Substituting in the inequalities:
It is true