Mason is working two summer jobs, making $7 per hour babysitting and making $10 per hour clearing tables. In a given week, he can work at most 12 total hours and must earn no less than $90. If xx represents the number of hours babysitting and yy represents the number of hours clearing tables, write and solve a system of inequalities graphically and determine one possible solution.
Let's define the variables:
xx = number of hours babysitting
yy = number of hours clearing tables
From the given information, we can derive the following equations:
1. The total number of hours worked cannot exceed 12: xx + yy ≤ 12
2. The minimum amount earned should be $90: 7xx + 10yy ≥ 90
To solve this system of inequalities graphically, we can plot these inequalities on a graph:
Equation 1: xx + yy ≤ 12
To graph this equation, we can draw a boundary line xx + yy = 12, which represents the maximum number of hours worked. To determine which side of the line to shade, we will test a point not on the line, such as (0,0).
0 + 0 ≤ 12
0 ≤ 12 (true)
Since (0,0) satisfies the inequality, the shaded region will be below and to the left of the line.
Equation 2: 7xx + 10yy ≥ 90
We can graph this equation by drawing a boundary line 7xx + 10yy = 90, which represents the minimum amount earned. Again, we will test a point not on the line, such as (0,0).
7(0) + 10(0) ≥ 90
0 ≥ 90 (false)
Since (0,0) does not satisfy the inequality, the shaded region will be above and to the right of the line.
Now let's find the intersection of the shaded region for both equations. The possible solution will be a point within this region.
Let's assume xx = 6 and yy = 6:
7(6) + 10(6) = 90
This solution satisfies both inequalities.
Therefore, one possible solution is xx = 6 hours babysitting and yy = 6 hours clearing tables.
Let xx represent the number of hours babysitting and yy represent the number of hours clearing tables.
According to the problem, the following conditions must be satisfied:
1) $7xx + $10yy ≥ $90 (minimum amount of money earned)
2) xx + yy ≤ 12 (maximum total number of hours worked)
In order to graphically represent these inequalities, we first need to solve each inequality for yy.
1) $7xx + $10yy ≥ $90
Solving for yy: $10yy ≥ $90 - $7xx
yy ≥ ($90 - $7xx)/$10
2) xx + yy ≤ 12
Solving for yy: yy ≤ 12 - xx
Now we can graph each inequality separately.
For the first inequality, we have:
yy ≥ ($90 - $7xx)/$10
Let's graph this inequality:
First, plot the line y = ($90 - $7xx)/$10:
- When xx = 0, yy = ($90 - $7(0))/$10 = $90/$10 = $9 (point: (0, 9))
- When xx = 10, yy = ($90 - $7(10))/$10 = $20/$10 = $2 (point: (10, 2))
Plotting these two points on a graph and drawing a line passing through them, we get:
```
y
|\
| \
| \
| \
| \
| \
| \
------------------ x
```
The line should be solid because yy is greater than or equal to ($90 - $7xx)/$10.
For the second inequality, we have:
yy ≤ 12 - xx
Let's graph this inequality:
First, plot the line y = 12 - xx:
- When xx = 0, yy = 12 - 0 = 12 (point: (0, 12))
- When xx = 10, yy = 12 - 10 = 2 (point: (10, 2))
Plotting these two points on a graph and drawing a line passing through them, we get:
```
y
| /
| /
| /
| /
| /
|/
------------------ x
```
The line should be dashed because yy is less than or equal to 12 - xx.
Now we can shade the region that satisfies both inequalities.
```
y
|\
| \
| \ /
| \ /
| \/
|
|
------------------ x
```
The shaded region represents the possible solutions.
One possible solution within this region could be: xx = 6 and yy = 6. This means Mason works 6 hours babysitting and 6 hours clearing tables.
To solve this problem, let's set up a system of inequalities based on the given information.
Let xx be the number of hours babysitting and yy be the number of hours clearing tables.
We know the following information:
1. The total number of hours worked cannot exceed 12: xx + yy ≤ 12
2. The minimum amount that Mason must earn is $90: 7xx + 10yy ≥ 90
Now, let's plot the above two inequalities on a graph to find one possible solution.
First, let's plot the first inequality, xx + yy ≤ 12. To do this, we need to find the line that represents this inequality when it becomes an equation: xx + yy = 12.
To plot the line, we can choose two different points that satisfy the equation. Let's choose (0, 12) and (12, 0). When we plot these points and draw a line through them, we will have the line xx + yy = 12.
Next, let's plot the second inequality, 7xx + 10yy ≥ 90. To do this, we need to find the line that represents this inequality when it becomes an equation: 7xx + 10yy = 90.
Again, we can choose two different points to plot the line. Let's choose (0, 9) and (10, 0). When we plot these points and draw a line through them, we will have the line 7xx + 10yy = 90.
Finally, we need to shade the feasible region, which is the region on the graph that satisfies both inequalities.
Now, let's find one possible solution by finding the coordinates of a point within the shaded region. This point represents a combination of hours babysitting (xx) and hours clearing tables (yy) that satisfies both inequalities.
Once the graph is drawn and shaded, you can determine a coordinate within the shaded region, such as (6, 6). This means that Mason can work 6 hours babysitting and 6 hours clearing tables, which satisfies both inequalities: xx + yy ≤ 12 and 7xx + 10yy ≥ 90.
Therefore, one possible solution is xx = 6 and yy = 6, which means that Mason can work 6 hours babysitting and 6 hours clearing tables to meet the given conditions.