What is the slope intercept equation for the line containing the points (3, 2) and (2, 3)?
Understand the Problem
The question is asking to find the slope-intercept equation of a line that passes through two given points, (3, 2) and (2, 3). This involves calculating the slope using the points and then using the slope to formulate the equation in slope-intercept form (y = mx + b).
Answer
The equation is \( y = -x + 5 \).
Answer for screen readers
The slope-intercept equation for the line is ( y = -x + 5 ).
Steps to Solve
- Calculate the Slope
To find the slope ( m ) between the points ( (3, 2) ) and ( (2, 3) ), use the formula:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Substituting the coordinates:
$$ m = \frac{3 - 2}{2 - 3} = \frac{1}{-1} = -1 $$
- Use the point-slope form
Now that we have the slope, we can use the point-slope form of the equation:
$$ y - y_1 = m(x - x_1) $$
Using point ( (3, 2) ):
$$ y - 2 = -1(x - 3) $$
- Simplify to slope-intercept form
Distributing and simplifying the equation:
$$ y - 2 = -x + 3 $$
Adding 2 to both sides:
$$ y = -x + 5 $$
- Final equation
The slope-intercept equation for the line is:
$$ y = -x + 5 $$
The slope-intercept equation for the line is ( y = -x + 5 ).
More Information
This equation indicates that the line has a slope of -1 and a y-intercept of 5. The negative slope means that as ( x ) increases, ( y ) decreases.
Tips
- Confusing the coordinates when calculating the slope. Always double-check that ( y_1 ) and ( y_2 ) correspond correctly to ( x_1 ) and ( x_2 ).
- Forgetting to simplify the equation to the slope-intercept form.
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