Write the equation of the line graphed below in simplest form.
Understand the Problem
The question is asking for the equation of a line based on a graph. To solve this, we need to identify key characteristics of the line, such as its slope and y-intercept, and then formulate the equation in the slope-intercept form (y = mx + b).
Answer
The equation of the line is \( y = -x + 2 \).
Answer for screen readers
The equation of the line is ( y = -x + 2 ).
Steps to Solve
- Identify two points on the line
Find two points where the line crosses the grid intersections. For this line, let's assume we take points ((-8, 6)) and ((2, -4)).
- Calculate the slope (m)
The slope (m) can be calculated using the formula:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Substituting in our points:
$$ m = \frac{-4 - 6}{2 - (-8)} = \frac{-10}{10} = -1 $$
- Identify the y-intercept (b)
From the graph, the line crosses the y-axis at the point ((0, 2)). Therefore, the y-intercept (b = 2).
- Write the slope-intercept form
The slope-intercept form is given by the equation:
$$ y = mx + b $$
Substituting in our values for (m) and (b):
$$ y = -1x + 2 $$
- Simplify the equation
The final equation of the line can be written in standard form as:
$$ y = -x + 2 $$
The equation of the line is ( y = -x + 2 ).
More Information
This line has a negative slope, which indicates that it decreases as (x) increases. The y-intercept shows that when (x=0), (y=2).
Tips
- Forgetting to calculate the slope correctly. Always ensure you subtract the correct coordinates.
- Misreading the y-intercept from the graph. Confirm the point where the line crosses the y-axis.
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