To determine the equation of a line that passes through the points [tex]\((-2, -5)\)[/tex] and [tex]\( (6, 1)\)[/tex], we will use the point-slope form of a linear equation:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept. Here is the step-by-step process:
### Step 1: Calculate the slope (m)
The slope [tex]\( m \)[/tex] 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]
Plugging in the coordinates of our points [tex]\((x_1, y_1) = (-2, -5)\)[/tex] and [tex]\((x_2, y_2) = (6, 1)\)[/tex], we get:
[tex]\[ m = \frac{1 - (-5)}{6 - (-2)} = \frac{1 + 5}{6 + 2} = \frac{6}{8} = \frac{3}{4} \][/tex]
Thus, the slope [tex]\( m \)[/tex] of our line is [tex]\( 0.75 \)[/tex].
### Step 2: Calculate the y-intercept (b)
Using the slope-intercept form [tex]\( y = mx + b \)[/tex] and one of the given points, we can solve for [tex]\( b \)[/tex]. Let's use point [tex]\((-2, -5)\)[/tex]:
[tex]\[ y = mx + b \][/tex]
Plug in [tex]\( m = 0.75 \)[/tex], [tex]\( x = -2 \)[/tex], and [tex]\( y = -5 \)[/tex]:
[tex]\[ -5 = 0.75 \cdot (-2) + b \][/tex]
[tex]\[ -5 = -1.5 + b \][/tex]
[tex]\[ b = -5 + 1.5 \][/tex]
[tex]\[ b = -3.5 \][/tex]
Thus, the y-intercept [tex]\( b \)[/tex] is [tex]\(-3.5\)[/tex].
### Step 3: Write the equation of the line
Now that we have the slope [tex]\( m = 0.75 \)[/tex] and the y-intercept [tex]\( b = -3.5 \)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[ y = 0.75x - 3.5 \][/tex]
So, the equation of the line that passes through the points [tex]\((-2, -5)\)[/tex] and [tex]\( (6, 1) \)[/tex] is:
[tex]\[ y = 0.75x - 3.5 \][/tex]
