Problem 29: Write an equation for the line in point-slope form and in standard form.
line through (2, 3) and (3, 5)
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To find the equation of the line through the points (2,3) and (3,5), we will first find the slope using the formula:
slope = (y2 - y1) / (x2 - x1)
slope = (5 - 3) / (3 - 2) = 2 / 1 = 2
Now that we have the slope, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is a point on the line. Let's use the point (2,3):
y - 3 = 2(x - 2)
Now we can simplify:
y - 3 = 2x - 4
y = 2x - 1
So, the equation of the line in point-slope form is y = 2x - 1.
To convert this into standard form (Ax + By = C), we simply move all terms to one side of the equation:
-2x + y = -1
Therefore, the equation of the line is -2x + y = -1 in standard form.