To determine the equations representing the proportional relationships between earnings and hours worked for both employees, we need to identify the relationship for each one based on the data provided.
Employee 1:
From the table, we can see the following information:
- \(1\) hour → \($10\)
- \(2\) hours → \($20\)
- \(3\) hours → \($30\)
To derive the equation, we notice that for Employee 1, the earnings (\(E\)) are directly proportional to the hours (\(H\)) worked.
The ratio of earnings to hours can be calculated as follows:
- For \(1\) hour: \(\frac{10}{1} = 10\)
- For \(2\) hours: \(\frac{20}{2} = 10\)
- For \(3\) hours: \(\frac{30}{3} = 10\)
This indicates a consistent relationship where the earnings per hour is \($10\). Therefore, the equation that represents this relationship can be expressed as:
\[ E_1 = 10H \]
Where \(E_1\) is the earnings of Employee 1 and \(H\) is the number of hours worked.
Employee 2:
For Employee 2, the points given are:
- \((0, 0)\)
- \((1, 15)\)
- \((3, 45)\)
- \((5, 75)\)
From these points, we can determine the relationship. The earnings for Employee 2 can be derived from the points given. The earnings per hour can be calculated as follows for each point:
- From \((1, 15)\): \(\frac{15}{1} = 15\)
- From \((3, 45)\): \(\frac{45}{3} = 15\)
- From \((5, 75)\): \(\frac{75}{5} = 15\)
This shows that the earnings per hour is consistently \($15\).
Thus, the equation that represents this relationship is:
\[ E_2 = 15H \]
Where \(E_2\) is the earnings of Employee 2, and \(H\) is the number of hours worked.
Summary:
- For Employee 1: \(E_1 = 10H\)
- For Employee 2: \(E_2 = 15H\)
These equations describe the proportional relationships between earnings and hours worked for each employee.
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