To find the equation of the line that passes through the points [tex]\((0, 3)\)[/tex] and [tex]\((7, 0)\)[/tex], we need to determine the slope (m) and the y-intercept (b) of the line in the slope-intercept form [tex]\( y = mx + b \)[/tex].
### Step 1: Calculate the Slope (m)
The slope (m) of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((0, 3)\)[/tex] and [tex]\((7, 0)\)[/tex]:
[tex]\[ (x_1, y_1) = (0, 3) \][/tex]
[tex]\[ (x_2, y_2) = (7, 0) \][/tex]
So, we plug in the coordinates:
[tex]\[ m = \frac{0 - 3}{7 - 0} = \frac{-3}{7} \][/tex]
The slope of the line is [tex]\( m = -\frac{3}{7} \)[/tex].
### Step 2: Find the Y-Intercept (b)
To find the y-intercept (b), we use the slope-intercept form equation [tex]\( y = mx + b \)[/tex] and substitute one of the points into the equation. We can use the point [tex]\((0, 3)\)[/tex]:
[tex]\[ y = mx + b \][/tex]
[tex]\[ 3 = -\frac{3}{7}(0) + b \][/tex]
[tex]\[ 3 = b \][/tex]
Thus, the y-intercept (b) is 3.
### Step 3: Write the Equation of the Line
Now that we have the slope and y-intercept, we can write the equation of the line in the form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = -\frac{3}{7}x + 3 \][/tex]
### Summary
The equation of the line that passes through the points [tex]\((0, 3)\)[/tex] and [tex]\((7, 0)\)[/tex] is:
[tex]\[ y = -\frac{3}{7}x + 3 \][/tex]
Among the given options, the correct answer is:
[tex]\[ \boxed{y = -\frac{3}{7} x + 3} \][/tex]
