The instructor had saved $3,500 and rents an apartment for $275 monthly. He believes the point (6, 1850) would be on the equation of the line. Is he correct? Explain how to check if a point is on a line.

Answer 1

To determine whether the point (6, 1850) is on the line represented by the instructor's savings and expenses, we first need to establish the equation of the line based on the given information.

1. Identify the variables:

   • Let's denote \( x \) as the number of months and \( y \) as the amount of money saved after \( x \) months.
   • The instructor starts with $3,500 and pays $275 each month.

2. Formulate the equation:

   • The amount of savings after \( x \) months can be represented by the equation: \[ y = 3500 - 275x \]
   • This equation indicates that starting from an initial amount of $3,500, the instructor's savings decrease by $275 for every month (x).

3. Check if the point (6, 1850) lies on the line:

   • Substitute \( x = 6 \) into the equation to find \( y \): \[ y = 3500 - 275(6) \]
   • Compute \( 275 \times 6 \): \[ 275 \times 6 = 1650 \]
   • Now, calculate \( y \): \[ y = 3500 - 1650 = 1850 \]

4. Conclusion:

   • The calculated \( y \) value when \( x = 6 \) is 1850.
   • Therefore, the point (6, 1850) lies on the line described by the equation \( y = 3500 - 275x \), and the instructor is indeed correct.

How to Check if a Point is on a Line:

To check if a point \((a, b)\) lies on the line defined by an equation \( y = f(x) \):

1. Substitute \( a \) (the x-coordinate of the point) into the equation to calculate \( f(a) \).
2. Compare the calculated value \( f(a) \) with \( b \) (the y-coordinate of the point).
3. If \( f(a) = b \), then the point \((a, b)\) lies on the line.
4. If \( f(a) \neq b \), then the point does not lie on the line.