Equations of Lines Derivations

Transcription

1 Equations of Lines Derivations If you know how slope is defined mathematically, then deriving equations of lines is relatively simple. We will start off with the equation for slope, normally designated by the letter m, and derive the Point Slope form of a line, Slope Intercept, and General Form of an Equation of a Line. Slope y 2 y 1 x 2 x 1 = m Using the slope formula, I will substitute x for x 2 and y for y 2, that results in y y 1 x x 1 = m Multiply both sides by CD; x x 1; y y 1 = m(x x 1 ) That simple multiplication by the denominator in the slope formula yields y y 1 = m(x x 1 ) which is the Point Slope Form of a Line The Point Slope Form of a Line is generally used for writing/finding an equation of a line. Example 1 Find an equation of a line that passes through (2, 5) with slope 3. Using y y 1 = m(x x 1 ), substitute the values of x and y from the ordered pair and the value of the slope into the equation. y ( 5) = 3(x 2) this is an answer, but I could simplify further y + 5 = 3(x + 2) y + 5 = 3x + 6 y = 3x + 1 Example 2 Find an equation of a line that passes through (2, 1) and (4, 9). Using y y 1 = m(x x 1 ), I can substitute the values of x and y from either ordered pair. The slope was not given, but I can find it by the slope formula. m = = 10 2 = 5 Using the ordered pair (2, 1) and substituting 1

2 y 1 = 5(x 2) y 1 = 5x + 10 y = 5x + 11 If I did enough problems like examples 1 & 2 and simplified them by solving for y, we would see that always results in an equivalent equation the form of y = mx + b. The b is the result of using the Distributive Property and combining like terms. In both cases you can see the slope is the coefficient of the x. Interestingly, when solving the Point Slope Form of an Equation of a Line for y, it always results in an equation of the form y = mx + b. The y-intercept of a graph, where a graph crosses the y-axis, occurs when the value of x is 0. In the equation y = mx + b, when x = 0, then y = b. So b is called the y-intercept and it is where the graph crosses the y=axis. y = mx + b is called the Slope Intercept Form of a Line The Slope Intercept Form of a Line is normally used for graphing. The graph of a linear equation is a line - a line is defined by two points. If one point is given to us, the y- intercept, we can find another point by using the slope. Example 3 Graph y = 3x + 2 The y-intercept is 2, located at (0, 2) The slope is 3 or 3/1. That means from (0,2), I go up 3 and over 1 to find another point. Example 4 Graph y = 2 3 x + 1 The y-intercept is 1, located at (0, 1) The slope is 2/3. That means from (0, 1), I go down 2 and over 3. OR I could have gone up 2 and over 3 from (0, 1) 2

3 By knowing the Slope Intercept Form of an Equation of a Line, I can graph it by inspection rather than making an x-y chart and plugging in points. Again and again we see, the more math you know, the easier it gets. Example 5 Find the slope and y-intercept of y = 4x 2. By inspection, the slope is 4 or 4/1 and the y-intercept occurs at (0, 2) If I were to graph that equation, I would place a point on (0, 2), then from there count up 4 and go over 1 and place another point. Drawing a line through those two points is the graph of the equation. Now if I took an equation, such as in Examle 5, y = 4x 2 and placed all the variables on one side and the constant on the other side of the equation, y = 4x 2 would look like 4x y = 2 An equation in the form, Ax + By = C is said to be the General Form of an Equation of a Line. Equations are often written in this form. The question is, how much information can I gleam from an equation written in General Form? Well, we already now that the intercepts occur when the other variable is zero. That is, the y-intercept occurs when x = 0 and the x-intercept occurs when y = 0. So by looking at an equation in General Form, we can readily identify where the graph crosses the x and y axes. Example 6 Find the x and y intercepts of 2x + 5y = 10 The y-intercept occurs when x = 0, if x = 0, then 5y = 10 or y int = 2. The x-intercept occurs when y = 0, if y = 0, then 2x = 10 or x int = 5 The x int occurs at (5, 0) and the y int occurs at (0, 2) Example 7 Find the x and y intercept of 3x 2y = 6 If x = 0, then 2y = 6 or y int = 3 (0, 3) is the y int If y = 0, then 3x = 6, or x int = 2 (2, 0) is the x int In examples 6 and 7, those two intercepts could be plotted and a line draw through those two points for the graph of the line. 3

4 If we looked further at those two equations, we might be able to find more information by inspection. Lets look at the last two equations written in General Form and solve the for y, placing them in Slope Intercept Form. 2x + 5y = 10 3x 2y = 6 5y = 2x y = 3x + 6 y = 2 5 x + 2 y = 3 2 x 3 In these two problems, the y-intercepts are the same as we found using the General Form and since the equations are now in Slope Intercept Form, we can see the slopes are 2 5 and 3 2 respectively. Looking at the two equations in General Form; 2x + 5y = 10 and 3x 2y = 6, is there a pattern that might suggest I could find the slopes of those two lines without solving for y? Let s see 2x + 5y = 10, slope = 2 5 and 3x 2y = 6, slope = 3 2 It appears if I put the coefficient of the x over the coefficient of the y and take the opposite sign, that s the slope. That observation suggests that if we have an equation in General Form, in Ax + By = C, the slope = A B Let s show that more formally Solving for y, we have Ax + By = C By = Ax + C y = A B x + C B We can see that when we solve the General Form of an equation of a line for y, the coefficient of x will always be A, the slope. B Example 8 Find the x and y intercepts and the slope of 4x + 3y = 12 by inspection. x int occurs when y = 0, 4x = 12, x int = 3 y int occurs when x = 0, 3y = 12, y int = 4 4

5 The slope is A B, 4 3 Example 9 Find the x int, y int, slope and graph 5x 3y = 15 By inspection, the x int = 3, the y int = 5, and m = 5 3 = 5 3 Summary We can see all these equations of lines are related and came directly from the definition of slope. The Point Slope form of a line came directly from the slope formula. The Slope Intercept form of a line came directly from the Point Slope (solving for y), and the General Form of an equation of a line came from placing the x and y terms on the same side of the equation. Knowing those formulas allows us to find the x int, y int,, slope and graph by inspection. 5