The point-slope form of a line is y minus y one equals m times x minus x one. This form represents a line passing through a specific point x one, y one with a given slope m. In our case, we'll be working with the point (1, 2).
Slope describes the steepness and direction of a line. A positive slope means the line rises from left to right. A negative slope means it falls from left to right. A zero slope creates a horizontal line, and an undefined slope creates a vertical line. All these lines can pass through our point (1, 2).
Our given point is (1, 2). This means x one equals 1 and y one equals 2. This point will serve as our anchor for any line equation we write. All lines we create will pass through this specific point, making it a fundamental part of our line equations.
Let's see how different slopes affect the line through (1, 2). For slope 3, we substitute into point-slope form to get y minus 2 equals 3 times x minus 1, which simplifies to y equals 3x minus 1. For slope negative one-half, we get y minus 2 equals negative one-half times x minus 1, simplifying to y equals negative one-half x plus five-halves. For slope zero, we get y equals 2, a horizontal line.
When given two points, we first calculate the slope using the formula m equals y two minus y one over x two minus x one. For points (1, 2) and (3, 8), the slope is 8 minus 2 over 3 minus 1, which equals 6 over 2 or 3. Then we use point-slope form with either point to get y minus 2 equals 3 times x minus 1, which simplifies to y equals 3x minus 1.
Parallel lines have equal slopes. Given the line y equals 2x plus 5 with slope 2, to find a line through (1, 2) parallel to it, we use the same slope of 2. Substituting into point-slope form: y minus 2 equals 2 times x minus 1, which simplifies to y equals 2x. Both lines have the same slope but different y-intercepts.
Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is negative one. Given the line y equals 3x minus 1 with slope 3, the perpendicular slope is negative one-third. To find the line through (1, 2) perpendicular to the given line, we substitute into point-slope form: y minus 2 equals negative one-third times x minus 1, which simplifies to y equals negative one-third x plus seven-thirds.
The same line can be expressed in multiple forms. Starting with point-slope form y minus 2 equals 3 times x minus 1, we can convert to slope-intercept form by distributing and adding 2 to both sides, giving us y equals 3x minus 1. Rearranging terms gives us the standard form 3x minus y equals 1. All three equations represent the exact same line passing through (1, 2) with slope 3.