What is the equation of the line shown in the graph?
Understand the Problem
The question asks to determine the equation of the line from the given plot. We need to find the slope and y-intercept of the line and match it to the correct equation from the options.
Answer
$y = -\frac{2}{5}x - 3$
Answer for screen readers
$y = -\frac{2}{5}x - 3$
Steps to Solve
- Identify the y-intercept
The y-intercept is the point where the line crosses the y-axis. From the graph, the line crosses the y-axis at $y = -3$. So, the y-intercept is -3.
- Find two points on the line
Choose two points on the line to calculate the slope. Let's pick the points $(0, -3)$ and $(5, -5)$.
- Calculate the slope
The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Using the points $(0, -3)$ and $(5, -5)$, we have:
$m = \frac{-5 - (-3)}{5 - 0} = \frac{-5 + 3}{5} = \frac{-2}{5}$
So, the slope is $-\frac{2}{5}$.
- Write the equation of the line
The equation of a line in slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. We found that the slope is $-\frac{2}{5}$ and the y-intercept is $-3$. Therefore, the equation of the line is:
$y = -\frac{2}{5}x - 3$
$y = -\frac{2}{5}x - 3$
More Information
The slope-intercept form of a line is a convenient way to represent a linear equation, as it directly reveals the slope and y-intercept of the line. This makes it easy to visualize and analyze the line's behavior.
Tips
A common mistake is misreading the graph and incorrectly identifying the y-intercept or calculating the slope with incorrect coordinates or by swapping $x$ and $y$ values. Another common mistake is getting the sign of the slope wrong.
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