To find the equation of the line in slope-intercept form [tex]\(y = mx + b\)[/tex] that passes through the points [tex]\((2, 3)\)[/tex] and [tex]\((5, 7)\)[/tex], follow these steps:
1. Calculate the slope [tex]\(m\)[/tex]:
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 the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the given points [tex]\((2, 3)\)[/tex] and [tex]\((5, 7)\)[/tex]:
[tex]\[
m = \frac{7 - 3}{5 - 2} = \frac{4}{3}
\][/tex]
2. Find the y-intercept [tex]\(b\)[/tex]:
The y-intercept [tex]\(b\)[/tex] can be found using the slope-intercept form [tex]\(y = mx + b\)[/tex]. We can substitute one of the given points into the equation to solve for [tex]\(b\)[/tex]. Let's use the point [tex]\((2, 3)\)[/tex]:
[tex]\[
3 = \left(\frac{4}{3}\right) \cdot 2 + b
\][/tex]
Simplify the equation:
[tex]\[
3 = \frac{8}{3} + b
\][/tex]
To isolate [tex]\(b\)[/tex], subtract [tex]\(\frac{8}{3}\)[/tex] from both sides:
[tex]\[
3 - \frac{8}{3} = b
\][/tex]
Converting 3 to a fraction with a denominator of 3:
[tex]\[
\frac{9}{3} - \frac{8}{3} = b \quad \Rightarrow \quad \frac{1}{3} = b
\][/tex]
So, [tex]\(b = \frac{1}{3}\)[/tex].
3. Write the equation of the line:
Now that we have the slope [tex]\(m = \frac{4}{3}\)[/tex] and the y-intercept [tex]\(b = \frac{1}{3}\)[/tex], we can write the equation of the line in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y = \frac{4}{3}x + \frac{1}{3}
\][/tex]
Therefore, the equation of the line passing through the points [tex]\((2, 3)\)[/tex] and [tex]\((5, 7)\)[/tex] is:
[tex]\[
y = \frac{4}{3}x + \frac{1}{3}
\][/tex]
