To find the equation of the line that passes through the points [tex]\((-2, 11)\)[/tex] and [tex]\((9, 22)\)[/tex], we need to determine the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] of the line.
Step 1: Calculate the Slope [tex]\( m \)[/tex]
The slope 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]
Here, our points are [tex]\((-2, 11)\)[/tex] and [tex]\((9, 22)\)[/tex].
Substitute the coordinates into the slope formula:
[tex]\[ m = \frac{22 - 11}{9 - (-2)} \][/tex]
[tex]\[ m = \frac{22 - 11}{9 + 2} \][/tex]
[tex]\[ m = \frac{11}{11} \][/tex]
[tex]\[ m = 1 \][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\( 1 \)[/tex].
Step 2: Calculate the Y-intercept [tex]\( b \)[/tex]
The y-intercept [tex]\( b \)[/tex] of a line can be found using the equation of the line in point-slope form, converted to slope-intercept form:
[tex]\[ y = mx + b \][/tex]
We can use any of the given points to find [tex]\( b \)[/tex]. Let’s use the point [tex]\((-2, 11)\)[/tex]:
We know:
[tex]\[ y = 11 \][/tex]
[tex]\[ m = 1 \][/tex]
[tex]\[ x = -2 \][/tex]
Substitute these values into the equation [tex]\( y = mx + b \)[/tex]:
[tex]\[ 11 = 1 \cdot (-2) + b \][/tex]
[tex]\[ 11 = -2 + b \][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[ b = 11 + 2 \][/tex]
[tex]\[ b = 13 \][/tex]
So, the y-intercept [tex]\( b \)[/tex] is [tex]\( 13 \)[/tex].
Step 3: Write the Equation of the Line
Using the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex], the equation of the line is:
[tex]\[ y = mx + b \][/tex]
[tex]\[ y = 1x + 13 \][/tex]
[tex]\[ y = x + 13 \][/tex]
Therefore, the equation of the line is:
[tex]\[ y = x + 13 \][/tex]
