Question

graph the line with slope \\(\frac{2}{5}\\) and \\(y\\)-intercept \\(-5\\).

Explanation:

Step1: Recall the slope - intercept form

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. Given that the slope $m=\frac{2}{5}$ and the $y$-intercept $b=- 5$, the equation of the line is $y=\frac{2}{5}x-5$.

Step2: Plot the $y$-intercept

The $y$-intercept is $-5$, so we first plot the point $(0,-5)$ on the $y$-axis.

Step3: Use the slope to find another point

The slope $m = \frac{2}{5}$ can be thought of as $\frac{\text{rise}}{\text{run}}=\frac{2}{5}$. This means from the point $(0, - 5)$, we move up 2 units (rise) and then move to the right 5 units (run). So starting from $(0,-5)$, moving up 2 units gives us a $y$-coordinate of $-5 + 2=-3$, and moving right 5 units gives us an $x$-coordinate of $0 + 5 = 5$. So the new point is $(5,-3)$. We can also move down 2 units and left 5 units from $(0,-5)$: moving down 2 units gives $y=-5-2=-7$ and moving left 5 units gives $x = 0-5=-5$, so the point $(-5,-7)$ is also on the line.

Step4: Draw the line

After plotting the points $(0,-5)$, $(5,-3)$ (or $(-5,-7)$), we draw a straight line passing through these points.

Answer:

To graph the line:

1. Plot the point \((0, - 5)\) (the \(y\)-intercept).
2. From \((0,-5)\), use the slope \(\frac{2}{5}\): move up 2 units and right 5 units to get the point \((5,-3)\) (or move down 2 units and left 5 units to get \((-5,-7)\)).
3. Draw a straight line through the plotted points.