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There are two forms of a linear equation, and you will need to be able to convert one into the other. Slope-Intercept Form We've been calling this the 'y =' form. It looks like this: y = mx+ b where m is the slope and b is the y intercept (0, b). Both can be positive or negative. Because this form tells you two things about the line, as well as whether it slants up or down (depending on the sign of m), it's the most useful form. It also let's you quickly graph the line without doing any calculations. Standard Form The equation looks like this: Ax + By + C = 0 whare A, B, and C are constants and integers. (Some textbooks state that A must also be positive. We will follow that convention. However, textbooks may also call Ax + By = C 'Standard form'. We don't use that convention, as it doesn't align with the definition of Standard form for other equations you will see in Math 30. We call Ax + By + C = 0 'Standard form'). While we don't know anything useful about the line when its equation is in this format, it does lend itself to easily finding the x and y intercepts. The format is also useful for other reasons, and as a result, you will frequently see linear equations in this form. How to Convert From One Form to the Other 1. Converting Point-Slope Form to Standard Form We won't show or explain all the steps example 1: y = 3x + 6 -3x + y - 6 = 0 3x - y + 6 = 0 (switch all signs to make A positive) example 2: y = -4x - 1 4x + y + 1 = 0 One step! example 3: y = 2/5x + 7 -2/5x + y - 7 = 0 2/5x - y + 7 = 0 2x - 5y + 35 = 0 (multiply all terms by 5 to remove fractions) example 4: y = 3/4x - 1/3 -3/4x + y + 1/3 = 0 3/4x - y - 1/3 = 0 9x -12y - 4 = 0 (multiply all terms by 12 to remove fractions) 2. Converting Standard Form to Point-Slope Form This is just solving for y. We'll just show one example. example 1: 2x + 5y - 15 = 0 5y = -2x + 15 y = -2/5x + 3 Sample Problem: Are these lines parallel or perpendicular, or neither? 2x - 3y + 12 = 0 3x + 2y + 2 = 0 In standard form we can't tell, so we'll convert both to 'y =' form. Equation 1: 2x - 3y + 12 = 0 -3y = -2x -12 y = 2/3x + 4 m = 2/3 Equation 2: 3x + 2y + 2 = 0 2y = -3x - 2 y = -3/2x - 1 m = -3/2 The slopes are negative reciprocals, so the lines are perpendicular. Linear Equations | Parallel and Perpendicular | Equation Forms | Finding Equations |