Answer :
To find the equation of the line passing through the points [tex]\(\left(\frac{2}{5}, \frac{18}{20}\right)\)[/tex] and [tex]\(\left(\frac{1}{3}, \frac{11}{12}\right)\)[/tex], we can use the slope-intercept form of a line, which is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
Here are the steps to find the equation:
1. Convert the coordinates to simpler fractions:
- [tex]\(\frac{18}{20}\)[/tex] simplifies to [tex]\(\frac{9}{10}\)[/tex].
- Keep the other fractions as they are: [tex]\(\left(\frac{2}{5}, \frac{9}{10}\right)\)[/tex] and [tex]\(\left(\frac{1}{3}, \frac{11}{12}\right)\)[/tex].
2. Calculate the slope [tex]\(m\)[/tex]:
The formula for the slope [tex]\(m\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the given values:
[tex]\[
m = \frac{\frac{11}{12} - \frac{9}{10}}{\frac{1}{3} - \frac{2}{5}}
\][/tex]
- First, find a common denominator and subtract the fractions for the numerators and denominators:
- For the numerators: [tex]\(\frac{11}{12} - \frac{9}{10} = \frac{110 - 108}{120} = \frac{2}{120} = \frac{1}{60}\)[/tex]
- For the denominators: [tex]\(\frac{1}{3} - \frac{2}{5} = \frac{5 - 6}{15} = \frac{-1}{15}\)[/tex]
So, the slope [tex]\(m\)[/tex] becomes:
[tex]\[
m = \frac{\frac{1}{60}}{\frac{-1}{15}} = -\frac{1}{4}
\][/tex]
3. Find the y-intercept [tex]\(b\)[/tex]:
Use the equation [tex]\(y = mx + b\)[/tex] with one of the points. We'll use [tex]\(\left(\frac{2}{5}, \frac{9}{10}\right)\)[/tex].
Substitute [tex]\(x = \frac{2}{5}\)[/tex], [tex]\(y = \frac{9}{10}\)[/tex], and [tex]\(m = -\frac{1}{4}\)[/tex] into the equation:
[tex]\[
\frac{9}{10} = -\frac{1}{4} \times \frac{2}{5} + b
\][/tex]
- Calculate the product [tex]\(-\frac{1}{4} \times \frac{2}{5} = -\frac{2}{20} = -\frac{1}{10}\)[/tex]
Now solve for [tex]\(b\)[/tex]:
[tex]\[
\frac{9}{10} = -\frac{1}{10} + b
\][/tex]
Add [tex]\(\frac{1}{10}\)[/tex] to both sides:
[tex]\[
b = \frac{9}{10} + \frac{1}{10} = 1
\][/tex]
4. Write the equation of the line:
The equation [tex]\(y = mx + b\)[/tex] with [tex]\(m = -\frac{1}{4}\)[/tex] and [tex]\(b = 1\)[/tex] is:
[tex]\[
y = -\frac{1}{4}x + 1
\][/tex]
Therefore, the equation of the line passing through the given points in slope-intercept form is [tex]\(y = -\frac{1}{4}x + 1\)[/tex].
Here are the steps to find the equation:
1. Convert the coordinates to simpler fractions:
- [tex]\(\frac{18}{20}\)[/tex] simplifies to [tex]\(\frac{9}{10}\)[/tex].
- Keep the other fractions as they are: [tex]\(\left(\frac{2}{5}, \frac{9}{10}\right)\)[/tex] and [tex]\(\left(\frac{1}{3}, \frac{11}{12}\right)\)[/tex].
2. Calculate the slope [tex]\(m\)[/tex]:
The formula for the slope [tex]\(m\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the given values:
[tex]\[
m = \frac{\frac{11}{12} - \frac{9}{10}}{\frac{1}{3} - \frac{2}{5}}
\][/tex]
- First, find a common denominator and subtract the fractions for the numerators and denominators:
- For the numerators: [tex]\(\frac{11}{12} - \frac{9}{10} = \frac{110 - 108}{120} = \frac{2}{120} = \frac{1}{60}\)[/tex]
- For the denominators: [tex]\(\frac{1}{3} - \frac{2}{5} = \frac{5 - 6}{15} = \frac{-1}{15}\)[/tex]
So, the slope [tex]\(m\)[/tex] becomes:
[tex]\[
m = \frac{\frac{1}{60}}{\frac{-1}{15}} = -\frac{1}{4}
\][/tex]
3. Find the y-intercept [tex]\(b\)[/tex]:
Use the equation [tex]\(y = mx + b\)[/tex] with one of the points. We'll use [tex]\(\left(\frac{2}{5}, \frac{9}{10}\right)\)[/tex].
Substitute [tex]\(x = \frac{2}{5}\)[/tex], [tex]\(y = \frac{9}{10}\)[/tex], and [tex]\(m = -\frac{1}{4}\)[/tex] into the equation:
[tex]\[
\frac{9}{10} = -\frac{1}{4} \times \frac{2}{5} + b
\][/tex]
- Calculate the product [tex]\(-\frac{1}{4} \times \frac{2}{5} = -\frac{2}{20} = -\frac{1}{10}\)[/tex]
Now solve for [tex]\(b\)[/tex]:
[tex]\[
\frac{9}{10} = -\frac{1}{10} + b
\][/tex]
Add [tex]\(\frac{1}{10}\)[/tex] to both sides:
[tex]\[
b = \frac{9}{10} + \frac{1}{10} = 1
\][/tex]
4. Write the equation of the line:
The equation [tex]\(y = mx + b\)[/tex] with [tex]\(m = -\frac{1}{4}\)[/tex] and [tex]\(b = 1\)[/tex] is:
[tex]\[
y = -\frac{1}{4}x + 1
\][/tex]
Therefore, the equation of the line passing through the given points in slope-intercept form is [tex]\(y = -\frac{1}{4}x + 1\)[/tex].
