Let x = the number of birds that Tim has. Let y = the number of dogs that Tim has.
Tim has more than twice as many dogs as he has birds.
Tim has more than 2 birds.
The total number of dogs and birds that Tim has is less than 11.
Set up an appropriate system of linear inequalities.
Graph the system. Shade the solution region.
How many birds and how many dogs does Tim have?
To set up the system of linear inequalities, we consider the given information:
1) "Tim has more than twice as many dogs as he has birds."
This can be expressed as: y > 2x
2) "Tim has more than 2 birds."
This can be expressed as: x > 2
3) "The total number of dogs and birds that Tim has is less than 11."
This can be expressed as: x + y < 11
To graph the system, we plot the equations on a coordinate plane and shade the appropriate regions:
First, we graph the line y = 2x as a dashed line, and shade the region above the line (since y > 2x).
Second, we graph the line x = 2 as a solid vertical line at x = 2, and shade the region to the right of the line (since x > 2).
Finally, we graph the line x + y = 11 as a solid line, and shade the region below the line (since x + y < 11).
The region where all three shaded regions overlap is the solution region.
To find the number of birds and dogs Tim has, we find the point of intersection in the solution region.
Estimated (x, y) values are (3, 7), which means Tim has 3 birds and 7 dogs.
To set up a system of linear inequalities, we need to represent the given information as inequalities.
Let's translate the given information into inequalities:
1. "Tim has more than twice as many dogs as he has birds."
This can be expressed as: y > 2x (since y represents the number of dogs and x represents the number of birds).
2. "Tim has more than 2 birds."
This can be expressed as: x > 2.
3. "The total number of dogs and birds that Tim has is less than 11."
This can be expressed as: x + y < 11.
So, the system of linear inequalities can be written as:
y > 2x,
x > 2,
x + y < 11.
Now, let's graph the system and shade the solution region.
To do that, draw a graph with x representing the number of birds on the x-axis and y representing the number of dogs on the y-axis.
First, graph the boundaries for each inequality:
1. y > 2x
To graph this line, start by plotting the point (0,0) on the graph (since it satisfies the inequality). Then, find another point by selecting any value for x and solving for y. For example, let's take x = 1:
y > 2(1)
y > 2
Choose y = 3 as a point on the line. Connect the two points to create a dashed line, as the inequality is "greater than."
2. x > 2
This is a vertical line passing through x = 2. Draw a dashed line to represent that the inequality is "greater than."
3. x + y < 11
To graph this line, start by plotting the point (0,11) on the graph. Then, find another point by selecting any value for x and solving for y. For example, let's take x = 1:
1 + y < 11
y < 10
Choose y = 9 as a point on the line. Connect the two points to create a dashed line, as the inequality is "less than."
Finally, shade the region that satisfies all three inequalities. The shaded region represents the possible combinations of x and y that meet all the given conditions.
To find the number of birds and dogs, identify the point of intersection within the shaded region on the graph. The x-coordinate of that point represents the number of birds, and the y-coordinate represents the number of dogs.