To graph the equation [tex]\(3x + 5y = 2\)[/tex], we first need to put it in the slope-intercept form, 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 step-by-step calculations to achieve that:
1. Start with the given equation:
[tex]\[
3x + 5y = 2
\][/tex]
2. Solve for [tex]\(y\)[/tex]:
[tex]\[
5y = -3x + 2
\][/tex]
3. Divide every term by 5 to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{3}{5}x + \frac{2}{5}
\][/tex]
From the simplified equation [tex]\(y = -\frac{3}{5}x + \frac{2}{5}\)[/tex], we can identify that:
- The slope [tex]\(m\)[/tex] is [tex]\(-\frac{3}{5}\)[/tex].
- The y-intercept [tex]\(b\)[/tex] is [tex]\(\frac{2}{5}\)[/tex].
To graph this equation:
1. Start by plotting the y-intercept [tex]\(\left( 0, \frac{2}{5} \right)\)[/tex] on the y-axis. This is the point where the line crosses the y-axis.
2. Use the slope to find another point on the graph. The slope [tex]\(-\frac{3}{5}\)[/tex] indicates that for every 5 units you move to the right along the x-axis, you move 3 units down along the y-axis (because the slope is negative).
Starting from the y-intercept [tex]\(\left( 0, \frac{2}{5} \right)\)[/tex]:
- Move 5 units to the right (positive direction on the x-axis).
- Move 3 units down (negative direction on the y-axis).
This will give you another point:
[tex]\[
(0 + 5, \frac{2}{5} - 3) = (5, \frac{2}{5} - \frac{15}{5}) = (5, -\frac{13}{5})
\][/tex]
3. Plot this second point [tex]\((5, -\frac{13}{5})\)[/tex] on the graph.
4. Draw a straight line through the two points [tex]\(\left( 0, \frac{2}{5} \right)\)[/tex] and [tex]\((5, -\frac{13}{5})\)[/tex].
This line represents the equation [tex]\(3x + 5y = 2\)[/tex].
By following these steps, the correct graph of the equation should exhibit a line crossing the y-axis at [tex]\(\frac{2}{5}\)[/tex] and having a downward slope of [tex]\(-\frac{3}{5}\)[/tex].
