2 Identifying a Linear Equation Ax + By = CThe exponent of each variable is 1.The variables are added or subtracted.A or B can equal zero.A > 0Besides 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.
3 Examples of linear equations 2x + 4y =86y = 3 – xx = 1-2a + b = 5Equation is in Ax + By =C formRewrite with both variables on left side … x + 6y =3B =0 … x + 0 y =1Multiply both sides of the equation by -1 … 2a – b = -5Multiply both sides of the equation by 3 … 4x –y =-21
4 Examples of Nonlinear Equations The following equations are NOT in the standard form of Ax + By =C:The exponent is 2There is a radical in the equationVariables are multipliedVariables are divided4x2 + y = 5xy + x = 5s/r + r = 3
5 x and y -interceptsThe 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.
6 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 = 62x = 6x = 3So (3, 0) is the x-intercept of the line.
7 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 = 60 + y = 6y = 6So (0, 6) is the y-intercept of the line.
8 To summarize…. To find the x-intercept, plug in 0 for y. To find the y-intercept, plug in 0 for x.
9 Find the x and y- intercepts of x = 4y – 5 Plug in x = 0x = 4y - 50 = 4y - 55 = 4y= y(0, ) is they-interceptx-intercept:Plug in y = 0x = 4y - 5x = 4(0) - 5x = 0 - 5x = -5(-5, 0) is thex-intercept
10 Find the x and y-intercepts of g(x) = -3x – 1* Plug in x = 0g(x) = -3(0) - 1g(x) = 0 - 1g(x) = -1(0, -1) is thex-interceptPlug in y = 0g(x) = -3x - 10 = -3x - 11 = -3x= x( , 0) is the*g(x) is the same as y
11 Find the x and y-intercepts of 6x - 3y =-18 x-interceptPlug in y = 06x - 3y = -186x -3(0) = -186x - 0 = -186x = -18x = -3(-3, 0) is they-interceptPlug in x = 06x -3y = -186(0) -3y = -180 - 3y = -18-3y = -18y = 6(0, 6) is the
12 Find the x and y-intercepts of x = 3 x-interceptPlug in y = 0.There is no y. Why?x = 3 is a vertical line so x always equals 3.(3, 0) is the x-intercept.y-interceptA vertical line never crosses the y-axis.There is no y-intercept.xy
13 Find the x and y-intercepts of y = -2 y = -2 is a horizontal lineso y always equals -2.(0,-2) is the y-intercept.x-interceptPlug in y = 0.y cannot = 0 becausey = -2.y = -2 is a horizontalline so it never crossesthe x-axis.There is no x-intercept.xy
14 Graphing Equations Example: Graph the equation -5x + y = 2 Solve for y first.-5x + y = 2 Add 5x to both sidesy = 5x + 2The 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.
16 Graphing Equations Graph 4x - 3y = 12 Solve for y first 4x - 3y =12 Subtract 4x from both sides-3y = -4x Divide by -3y = x Simplifyy = x – 4The 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.