To find the equation of the line that passes through the points [tex]\((3, 3)\)[/tex] and [tex]\((1, 10)\)[/tex] and write it in slope-intercept form (which is [tex]\( y = mx + b \)[/tex]), we need to follow these steps:
1. Calculate the Slope:
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 the two points [tex]\((x_1, y_1) = (3, 3)\)[/tex] and [tex]\((x_2, y_2) = (1, 10)\)[/tex]:
[tex]\[
m = \frac{10 - 3}{1 - 3}
\][/tex]
[tex]\[
m = \frac{7}{-2}
\][/tex]
[tex]\[
m = -3.5
\][/tex]
2. Find the y-Intercept:
The slope-intercept form of the line is [tex]\( y = mx + b \)[/tex]. To find [tex]\( b \)[/tex] (the y-intercept), we use one of the points. We can use either point; let's use [tex]\((3, 3)\)[/tex].
Rewrite the slope-intercept form equation:
[tex]\[
y = mx + b
\][/tex]
Substitute [tex]\( m = -3.5 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( y = 3 \)[/tex] into the equation:
[tex]\[
3 = (-3.5)(3) + b
\][/tex]
Solve for [tex]\( b \)[/tex]:
[tex]\[
3 = -10.5 + b
\][/tex]
[tex]\[
b = 3 + 10.5
\][/tex]
[tex]\[
b = 13.5
\][/tex]
3. Write the Equation of the Line:
Now that we have both the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex], we can write the equation of the line. Substituting [tex]\( m = -3.5 \)[/tex] and [tex]\( b = 13.5 \)[/tex] into the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = -3.5x + 13.5
\][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[
y = -3.5x + 13.5
\][/tex]
