Earn al least $150 a week. $10 per hour working for a farmer. $5 per hour babysitting. can work at most 25 hours per week. Write and graph a system of linear inequalities.
x = hrs farming
y = hrs babysitting
10x + 5y >= 150
x + y <= 25
Solve the system
To write a system of linear inequalities that models the given scenario, we need to consider the conditions and constraints stated:
Let's define the following variables:
x = number of hours worked for the farmer
y = number of hours babysitting
Now let's establish the inequalities:
1. First, we need to ensure that the total weekly earnings are at least $150:
The weekly earnings from working for the farmer are $10 per hour, so the earnings from this job would be 10x.
The weekly earnings from babysitting are $5 per hour, so the earnings from this job would be 5y.
To ensure that the total weekly earnings are at least $150, the following inequality can be written:
10x + 5y ≥ 150
2. Next, we need to consider the maximum number of hours we can work in a week. The combined total of hours worked for the farmer and babysitting cannot exceed 25 hours:
The combined number of hours worked for the farmer and babysitting cannot exceed 25 hours, so the following inequality can be written:
x + y ≤ 25
To graph this system of inequalities, we will plot the lines represented by each inequality and shade the region that satisfies both of the inequalities. The shaded region will represent the feasible solutions.
Here is how you can graph this system of linear inequalities:
1. Graph the boundary line for the first inequality: 10x + 5y = 150
- To do this, rewrite the equation in slope-intercept form: y = (150 - 10x) / 5
- Pick two x-values and solve for the corresponding y-values to find two points on the line.
- Plot these points and draw the line passing through them.
2. Graph the boundary line for the second inequality: x + y = 25
- Rewrite the equation in slope-intercept form: y = 25 - x
- Pick two x-values and solve for the corresponding y-values to find two points on the line.
- Plot these points and draw the line passing through them.
3. Shade the region that satisfies both inequalities.
- Determine which side of each line represents a valid solution by choosing a point on one side and testing it in the inequalities.
- Shade the region that satisfies both inequalities.
The shaded region will represent the feasible solutions where you can earn at least $150 per week while working a maximum of 25 hours.