Let's work through the problem step-by-step.
1. Identify the data points given in the table:
- When [tex]\( x = 4 \)[/tex], [tex]\( y = 955 \)[/tex]
- When [tex]\( x = 7 \)[/tex], [tex]\( y = 1660 \)[/tex]
- When [tex]\( x = 10 \)[/tex], [tex]\( y = 2365 \)[/tex]
2. Determine the linear relationship:
- The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Calculate the slope [tex]\( m \)[/tex]:
- The slope [tex]\( m \)[/tex] shows how much [tex]\( y \)[/tex] changes as [tex]\( x \)[/tex] changes, defined as:
[tex]\[
m = \frac{\Delta y}{\Delta x}
\][/tex]
4. Slope Calculation (using the data points):
- The difference in [tex]\( y \)[/tex] between the first and second points is [tex]\( 1660 - 955 = 705 \)[/tex], and the difference in [tex]\( x \)[/tex] is [tex]\( 7 - 4 = 3 \)[/tex].
Hence, the slope for the first interval is:
[tex]\[
m = \frac{705}{3} = 235
\][/tex]
- The difference in [tex]\( y \)[/tex] between the second and third points is [tex]\( 2365 - 1660 = 705 \)[/tex], and the difference in [tex]\( x \)[/tex] is [tex]\( 10 - 7 = 3 \)[/tex].
Hence, the slope for the second interval is:
[tex]\[
m = \frac{705}{3} = 235
\][/tex]
5. Determine the y-intercept [tex]\( b \)[/tex]:
- Using the slope [tex]\( m = 235 \)[/tex] and substituting one of the points (e.g., [tex]\( x = 4 \)[/tex] and [tex]\( y = 955 \)[/tex]):
[tex]\[
955 = 235 \cdot 4 + b
\][/tex]
Solve for [tex]\( b \)[/tex]:
[tex]\[
955 = 940 + b \implies b = 955 - 940 = 15
\][/tex]
6. Formulate the linear equation:
- Substituting [tex]\( m = 235 \)[/tex] and [tex]\( b = 15 \)[/tex] into the general form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = 235x + 15
\][/tex]
7. Verify against the given options:
- Option 1: [tex]\( y = 235x + 15 \)[/tex] matches the equation we determined.
- Option 2: [tex]\( y = 4x + 955 \)[/tex]
- Option 3: [tex]\( y = 15x + 235 \)[/tex]
- Option 4: [tex]\( y = 3x + 705 \)[/tex]
None of the other options match our calculated equation.
8. Conclusion:
- The correct equation is [tex]\( y = 235x + 15 \)[/tex].
So, the linear equation that represents the scenario is [tex]\( y = 235x + 15 \)[/tex].
