To write the equation of the line passing through the points [tex]\((-2,5)\)[/tex] and [tex]\((7,10)\)[/tex], we need to follow these steps:
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 given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates [tex]\((x_1, y_1) = (-2, 5)\)[/tex] and [tex]\((x_2, y_2) = (7, 10)\)[/tex], we get:
[tex]\[
m = \frac{10 - 5}{7 - (-2)} = \frac{5}{9} = 0.5555555555555556
\][/tex]
2. Calculate the y-intercept [tex]\( b \)[/tex]:
To find the y-intercept [tex]\( b \)[/tex] of the line, we use the point-slope form of the equation of a line, which is:
[tex]\[
y = mx + b
\][/tex]
Solving for [tex]\( b \)[/tex], we get:
[tex]\[
b = y - mx
\][/tex]
We can use any of the two given points to calculate [tex]\( b \)[/tex]. Let's use the point [tex]\((-2, 5)\)[/tex]:
[tex]\[
b = 5 - 0.5555555555555556 \times (-2) = 5 + 1.1111111111111112 = 6.111111111111111
\][/tex]
3. Write the equation of the line:
Now that we have the slope [tex]\( m = 0.5555555555555556 \)[/tex] and the y-intercept [tex]\( b = 6.111111111111111 \)[/tex], we can write the equation of the line in the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = 0.5555555555555556x + 6.111111111111111
\][/tex]
So, the equation of the line passing through the points [tex]\((-2,5)\)[/tex] and [tex]\((7,10)\)[/tex] is:
[tex]\[
y = 0.5555555555555556x + 6.111111111111111
\][/tex]
