2 2 Linear Equations Identifying a Linear Equation

2. 2 Linear Equations

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 2 x + 4 y =8 6 y = 3 – x x=1 -2 a + b = 5 Equation is in Ax + By =C form Rewrite with both variables on left side … x + 6 y =3 B =0 … x + 0 y =1 Multiply both sides of the equation by -1 … 2 a – b = -5 Multiply both sides of the equation by 3 … 4 x –y =-21

Examples of Nonlinear Equations The following equations are NOT in the standard form of Ax + By =C: 4 x 2 + y = 5 The exponent is 2 There is a radical in the equation xy + x = 5 s/r + r = 3 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 2 x + y = 6, we know that y must equal 0. What must x equal? ● Plug in 0 for y and simplify. 2 x + 0 = 6 2 x = 6 x=3 So (3, 0) is the x-intercept of the line. ●

Finding the y-intercept ● For the equation 2 x + 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 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 = 4 y – 5 ● ● ● x-intercept: Plug in y = 0 x = 4 y - 5 x = 4(0) - 5 x=0 -5 x = -5 (-5, 0) is the x-intercept ● ● ● y-intercept: Plug in x = 0 x = 4 y - 5 0 = 4 y - 5 5 = 4 y =y (0, ) is the y-intercept

Find the x and y-intercepts of g(x) = -3 x – 1* ● ● ● x-intercept Plug in y = 0 g(x) = -3 x - 1 0 = -3 x - 1 1 = -3 x =x ( , 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 6 x - 3 y =-18 ● ● ● x-intercept Plug in y = 0 6 x - 3 y = -18 6 x -3(0) = -18 6 x - 0 = -18 6 x = -18 x = -3 (-3, 0) is the x-intercept ● ● ● y-intercept Plug in x = 0 6 x -3 y = -18 6(0) -3 y = -18 0 - 3 y = -18 -3 y = -18 y=6 (0, 6) is the y-intercept

Find the x and y-intercepts of x = 3 ● x-intercept ● ● Plug in y = 0. ● There is no y. Why? x = 3 is a vertical line so x always equals 3. ● ● y-intercept A 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 ● y-intercept ● Plug in y = 0. ● y = -2 is a horizontal line y cannot = 0 because y = -2. ● y = -2 is a horizontal line so it never crosses the x-axis. ●There so y always equals -2. ● (0, -2) is the y-intercept. x is no x-intercept. y

Graphing Equations ● ● Example: Graph the equation -5 x + y = 2 Solve for y first. -5 x + y = 2 Add 5 x to both sides y = 5 x + 2 The equation y = 5 x + 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 = 5 x + 2 x y

Graphing Equations ● Solve for y first Graph 4 x - 3 y = 12 4 x - 3 y =12 Subtract 4 x from both sides -3 y = -4 x + 12 Divide by -3 y= ● x+ Simplify y= x– 4 The equation y = x - 4 is in slope-intercept form, y=mx+b. The y -intercept is -4 and the slope is. Graph the line on the coordinate plane.

Graphing Equations Graph y = x - 4 x y