Let's find the equation of the line that passes through the points [tex]\((0,3)\)[/tex] and [tex]\((-5,-2)\)[/tex]. The general form of the equation of a line is [tex]\( y = mx + b \)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
### Step 1: Calculate the Slope ([tex]\(m\)[/tex])
The formula for the slope ([tex]\(m\)[/tex]) between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting in the coordinates of the given points:
[tex]\[ (x_1, y_1) = (0, 3) \][/tex]
[tex]\[ (x_2, y_2) = (-5, -2) \][/tex]
So, the slope [tex]\(m\)[/tex] is:
[tex]\[ m = \frac{-2 - 3}{-5 - 0} = \frac{-5}{-5} = 1.0 \][/tex]
### Step 2: Calculate the Y-Intercept ([tex]\(b\)[/tex])
The y-intercept ([tex]\(b\)[/tex]) can be found using the point-slope form of the line equation, which is rearranged into the form [tex]\( y = mx + b \)[/tex]. We substitute one of the points and the slope into this equation. Let's use the point [tex]\((0, 3)\)[/tex]:
Since [tex]\( y = mx + b \)[/tex],
[tex]\[ 3 = 1.0 \cdot 0 + b \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ b = 3.0 \][/tex]
### Step 3: Write the Equation of the Line
Now that we have the slope ([tex]\(m = 1.0\)[/tex]) and the y-intercept ([tex]\(b = 3.0\)[/tex]), we can write the equation of the line as:
[tex]\[ y = 1.0x + 3.0 \][/tex]
### Conclusion
The equation of the line that passes through the points [tex]\((0,3)\)[/tex] and [tex]\((-5,-2)\)[/tex] is:
[tex]\[ y = 1.0x + 3.0 \][/tex]
