Question

Calculate the equation of the line through the points (1, 1), (4, 3), (10, 7) and (40, 27).

y=2/3x-1/3

y=3/2x+1/2

y=2/3x+1/3

y=3/2x-1/2

Answer 1

To find the equation of the line that passes through the points (1, 1), (4, 3), (10, 7), and (40, 27), we can first attempt to calculate the slope using a pair of points and then determine the equation in slope-intercept form \( y = mx + b \).

### Step 1: Calculate the slope (m)
The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's use the points (1, 1) and (4, 3):

\[
m = \frac{3 - 1}{4 - 1} = \frac{2}{3}
\]

### Step 2: Calculate the y-intercept (b)
Now that we have the slope, we can use one of the points to find the y-intercept \( b \). We can use the point (1, 1):

\[
y = mx + b \quad \Rightarrow \quad 1 = \frac{2}{3}(1) + b
\]

Solving for \( b \):

\[
1 = \frac{2}{3} + b \quad \Rightarrow \quad b = 1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}
\]

### Step 3: Write the equation of the line
Now we have the slope \( m = \frac{2}{3} \) and the y-intercept \( b = \frac{1}{3} \). Thus, the equation of the line is:

\[
y = \frac{2}{3}x + \frac{1}{3}
\]

### Step 4: Verify with the other points
We can verify that the other points also satisfy this line equation:

- For \( (10, 7) \):
\[
y = \frac{2}{3}(10) + \frac{1}{3} = \frac{20}{3} + \frac{1}{3} = \frac{21}{3} = 7
\]

- For \( (40, 27) \):
\[
y = \frac{2}{3}(40) + \frac{1}{3} = \frac{80}{3} + \frac{1}{3} = \frac{81}{3} = 27
\]

### Conclusion
All points satisfy the same line equation. Therefore, the equation of the line through the points (1, 1), (4, 3), (10, 7), and (40, 27) is:

\[
\boxed{y = \frac{2}{3}x + \frac{1}{3}}
\]