Linear Equations Ax + By = C Identifying a Linear Equation ● ● ● ● ● ● ● Ax + By = C The exponent of each variable is 1. The variables are added or subtracted. A or B can equal zero. A>0 Besides x and y, other commonly used variables are m and n, a and b, and r and s. There are no radicals in the equation. Every linear equation graphs as a line. Examples of linear equations Equation is in Ax + By =C form 2x + 4y =8 6y = 3 – x Rewrite with both variables on left side … x + 6y =3 x=1 B =0 … x + 0 y =1 -2a + b = 5 Multiply both sides of the equation by -1 … 2a – b = -5 4x y 7 3 Multiply both sides of the equation by 3 … 12x –3y =-21 Examples of Nonlinear Equations The following equations are NOT in the standard form of Ax + By =C: 4x2 + y = 5 x4 xy + x = 5 s/r + r = 3 The exponent is 2 There is a radical in the equation Variables are multiplied Variables are divided x and y -intercepts ● The x-intercept is the point where a line crosses the x-axis. The general form of the x-intercept is (x, 0). The y-coordinate will always be zero. ● The y-intercept is the point where a line crosses the y-axis. The general form of the y-intercept is (0, y). The x-coordinate will always be zero. Finding the x-intercept ● For the equation 2x + y = 6, we know that y must equal 0. What must x equal? ● Plug in 0 for y and simplify. 2x + 0 = 6 2x = 6 x=3 So (3, 0) is the x-intercept of the line. ● Finding the y-intercept ● For the equation 2x + y = 6, we know that x must equal 0. What must y equal? ● Plug in 0 for x and simplify. 2(0) + y = 6 0+y=6 y=6 So (0, 6) is the y-intercept of the line. ● To summarize…. ● To find the x-intercept, plug in 0 for y. ● To find the y-intercept, plug in 0 for x. Find the x and y- intercepts of x = 4y – 5 ● ● ● x-intercept: Plug in y = 0 x = 4y - 5 x = 4(0) - 5 x=0-5 x = -5 (-5, 0) is the x-intercept ● ● y-intercept: Plug in x = 0 x = 4y - 5 0 = 4y - 5 5 = 4y 5 =y 4 ● 5 (0, 4 ) is the y-intercept Find the x and y-intercepts of g(x) = -3x – 1* ● ● ● x-intercept Plug in y = 0 g(x) = -3x - 1 0 = -3x - 1 1 = -3x 1 =x 3 1 ( 3 , 0) is the x-intercept *g(x) is the same as y ● ● ● y-intercept Plug in x = 0 g(x) = -3(0) - 1 g(x) = 0 - 1 g(x) = -1 (0, -1) is the y-intercept Find the x and y-intercepts of 6x - 3y =-18 ● ● ● x-intercept Plug in y = 0 6x - 3y = -18 6x -3(0) = -18 6x - 0 = -18 6x = -18 x = -3 (-3, 0) is the x-intercept ● ● ● y-intercept Plug in x = 0 6x -3y = -18 6(0) -3y = -18 0 - 3y = -18 -3y = -18 y=6 (0, 6) is the y-intercept Find the x and y-intercepts of x = 3 ● x-intercept ● ● Plug in y = 0. ●A There is no y. Why? x = 3 is a vertical line so x always equals 3. ● ● y-intercept vertical line never crosses the y-axis. ● There is no y-intercept. (3, 0) is the x-intercept. x Find the x and y-intercepts of y = -2 ● x-intercept Plug in y = 0. y cannot = 0 because y = -2. ● y = -2 is a horizontal line so it never crosses the x-axis. ● ●There ● y-intercept ● y = -2 is a horizontal line so y always equals -2. ● (0,-2) is the y-intercept. x is no x-intercept. y Graphing Equations ● Example: Graph the equation -5x + y = 2 Solve for y first. -5x + y = 2 Add 5x to both sides y = 5x + 2 ● The equation y = 5x + 2 is in slope-intercept form, y = mx+b. The y-intercept is 2 and the slope is 5. Graph the line on the coordinate plane. Graphing Equations Graph y = 5x + 2 x y Graphing Equations Graph 4x - 3y = 12 ● Solve for y first 4x - 3y =12 Subtract 4x from both sides -3y = -4x + 12 Divide by -3 y -4 = -3 12 x + -3 4 y = 3x – 4 ● Simplify The equation y = 43 x - 4 is in slope-intercept form, y=mx+b. The y -intercept is -4 and the slope is 4 . 3 Graph the line on the coordinate plane. Graphing Equations Graph y = 4 3x -4 x y

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