Question

1. (2,6) and (4,3) \\(x_1,y_1\\) \\(x_2,y_2\\) \\(\frac{3 - 6}{4 - 2}=\frac{-3}{2}=m\\) \\(y = mx + b\\) \\(y = \frac{-3}{2}x + b\\) \\(6 = \frac{-3}{2}(2)+b\\) \\(6 = -3 + b\\) \\(6 + 3 = b\\) \\(9 = b\\) 3. (0,4) and (1,7) \\(x_1,y_1\\) \\(x_2,y_2\\) \\(y = mx + b\\)

Explanation:

Step1: Identify the two points

We have the points \((0, 4)\) and \((1, 7)\). Let \( (x_1, y_1)=(0, 4) \) and \( (x_2, y_2)=(1, 7) \).

Step2: Calculate the slope \( m \)

The formula for slope \( m \) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
Substituting the values, we get \( m=\frac{7 - 4}{1 - 0}=\frac{3}{1} = 3 \).

Step3: Find the y - intercept \( b \)

The equation of a line is \( y=mx + b \). We know that when \( x = 0 \), \( y = 4 \) (from the point \((0,4)\)). Substituting \( x = 0 \), \( y = 4 \) and \( m = 3 \) into the equation \( y=mx + b \), we get \( 4=3\times0 + b \), so \( b = 4 \).

Step4: Write the equation of the line

Using \( m = 3 \) and \( b = 4 \) in the equation \( y=mx + b \), we get \( y=3x + 4 \).

Answer:

The equation of the line passing through \((0, 4)\) and \((1, 7)\) is \( y = 3x+4 \)