2.2 – Linear Equations

2.2 – Linear EquationsLinear equation

2.2 – Linear EquationsLinear equation – equation with only addition,

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction,

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication,

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a

variable by a number.

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a

variable by a number.

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a

variable by a number.

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a

variable by a number.

Linear Eqs.

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a

variable by a number.

Linear Eqs.

5x – 3y = 7

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a

variable by a number.

Linear Eqs.

5x – 3y = 7

x = 9

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a

variable by a number.

Linear Eqs.

5x – 3y = 7

x = 9

6s = -3t – 15

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a

variable by a number.Linear Eqs. 5x – 3y = 7 x = 9 6s = -3t – 15 y = ½x

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a variable by a number.

Linear Eqs. Non-linear Eqs.5x – 3y = 7 x = 9 6s = -3t – 15 y = ½x

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a variable by a number.

Linear Eqs. Non-linear Eqs.5x – 3y = 77a + 4b2 = -8x = 9 6s = -3t – 15 y = ½x

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a variable by a number.

Linear Eqs. Non-linear Eqs.5x – 3y = 77a + 4b2 = -8x = 9 6s = -3t – 15 y = ½x

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a variable by a number.

Linear Eqs. Non-linear Eqs.5x – 3y = 77a + 4b2 = -8x = 9 y = √x + 56s = -3t – 15 y = ½x

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a variable by a number.

Linear Eqs. Non-linear Eqs.5x – 3y = 77a + 4b2 = -8x = 9 y = √x + 56s = -3t – 15 y = ½x

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a variable by a number.

Linear Eqs. Non-linear Eqs.5x – 3y = 77a + 4b2 = -8x = 9 y = √x + 56s = -3t – 15 x + xy = 1y = ½x

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a variable by a number.

Linear Eqs. Non-linear Eqs.5x – 3y = 77a + 4b2 = -8x = 9 y = √x + 56s = -3t – 15 x + xy = 1y = ½x

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a variable by

a number.Linear Eqs. Non-linear Eqs.5x – 3y = 7 7a + 4b2 = -8x = 9 y = √x + 56s = -3t – 15 x + xy = 1y = ½x y = 1

x

2.2 – Linear EquationsLinear equation – equation with only addition,

subtraction, multiplication, and division of a variable by

a number.Linear Eqs. Non-linear Eqs.5x – 3y = 7 7a + 4b2 = -8x = 9 y = √x + 56s = -3t – 15 x + xy = 1y = ½x y = 1

x

Example 1 State whether each function or equation is linear. If no, explain why.

Example 1 State whether each function or equation is linear. If no, explain why.

(a) f(x) = 10 – x

Example 1 State whether each function or equation is linear. If no, explain why.

(a) f(x) = 10 – x YES

Example 1 State whether each function or equation is linear. If no, explain why.

(a) f(x) = 10 – x YES

(b) g(x) = x4 – 5

Example 1 State whether each function or equation is linear. If no, explain why.

(a) f(x) = 10 – x YES

(b) g(x) = x4 – 5 NO

Example 1 State whether each function or equation is linear. If no, explain why.

(a) f(x) = 10 – x YES

(b) g(x) = x4 – 5 NO; exponent on var.

Example 1 State whether each function or equation is linear. If no, explain why.

(a) f(x) = 10 – x YES

(b) g(x) = x4 – 5 NO; exponent on var.

(c) h(x,y) = 2xy

Example 1 State whether each function or equation is linear. If no, explain why.

(a) f(x) = 10 – x YES

(b) g(x) = x4 – 5 NO; exponent on var.

(c) h(x,y) = 2xy NO

Example 1 State whether each function or equation is linear. If no, explain why.

(a) f(x) = 10 – x YES

(b) g(x) = x4 – 5 NO; exponent on var.

(c) h(x,y) = 2xy NO; multiplying vars.

• Standard Form

• Standard Form = Ax + By = C

• Standard Form = Ax + By = C

*Get x’s and y’s on left side,

• Standard Form = Ax + By = C

*Get x’s and y’s on left side, numbers on rt.

• Standard Form = Ax + By = C

*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and

C.

• Standard Form = Ax + By = C

*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and

C.

(a) y = -2x + 3

• Standard Form = Ax + By = C

*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and

C.

(a) y = -2x + 3

+2x +2x

• Standard Form = Ax + By = C

*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and

C.

(a) y = -2x + 3

+2x +2x

2x + y = 3

• Standard Form = Ax + By = C

*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and

C.

(a) y = -2x + 3

+2x +2x

2x + y = 3

A=2

• Standard Form = Ax + By = C

*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and

C.

(a) y = -2x + 3

+2x +2x

2x + y = 3

A=2,B=1

• Standard Form = Ax + By = C

*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and

C.

(a) y = -2x + 3

+2x +2x

2x + y = 3

A=2,B=1 ,&C=3

• Standard Form = Ax + By = C

*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and

C.

(a) y = -2x + 3 (b) ⅜x = 3y + 2

+2x +2x

2x + y = 3

A=2,B=1,&C=3

• Standard Form = Ax + By = C*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and C.

(a) y = -2x + 3 (b) ⅜x = 3y + 2 +2x +2x -3y -3y

2x + y = 3 A=2,B=1,&C=3

• Standard Form = Ax + By = C*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and

C.

(a) y = -2x + 3 (b) ⅜x = 3y + 2 +2x +2x -3y -3y

2x + y = 3 ⅜x – 3y = 2

A=2,B=1,&C=3

• Standard Form = Ax + By = C

*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and C.

(a) y = -2x + 3 (b) ⅜x = 3y + 2

+2x +2x -3y -3y

2x + y = 3 ⅜x – 3y = 2

A=2,B=1,&C=3 8(⅜x – 3y) = (2)8

• Standard Form = Ax + By = C*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and C.

(a) y = -2x + 3 (b) ⅜x = 3y + 2 +2x +2x -3y -3y

2x + y = 3 ⅜x – 3y = 2A=2,B=1,&C=3 8(⅜x – 3y) = (2)8

3x – 24y = 16

• Standard Form = Ax + By = C*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and C.

(a) y = -2x + 3 (b) ⅜x = 3y + 2 +2x +2x -3y -3y

2x + y = 3 ⅜x – 3y = 2A=2,B=1,&C=3 8(⅜x – 3y) = (2)8

3x – 24y = 16A=3

• Standard Form = Ax + By = C*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and C.

(a) y = -2x + 3 (b) ⅜x = 3y + 2 +2x +2x -3y -3y

2x + y = 3 ⅜x – 3y = 2A=2,B=1,&C=3 8(⅜x – 3y) = (2)8

3x – 24y = 16A=3,B=-24

• Standard Form = Ax + By = C*Get x’s and y’s on left side, numbers on rt.

Example 2 Write each equation in standard form. Identify A, B, and C.

(a) y = -2x + 3 (b) ⅜x = 3y + 2 +2x +2x -3y -3y

2x + y = 3 ⅜x – 3y = 2A=2,B=1,&C=3 8(⅜x – 3y) = (2)8

3x – 24y = 16A=3,B=-

24,&C=16

• x-intercept

• x-intercept – (x, 0)

• x-intercept – (x, 0); y-intercept

• x-intercept – (x, 0); y-intercept – (0, y)

• x-intercept – (x, 0); y-intercept – (0, y)

Example 3 Find the x and y intercepts of 3x – 4y – 12 = 0. Then graph the equation.

• x-intercept – (x, 0); y-intercept – (0, y)

Example 3 Find the x and y intercepts of 3x – 4y – 12 = 0. Then graph the equation.

3x – 4y – 12 = 0

• x-intercept – (x, 0); y-intercept – (0, y)

Example 3 Find the x and y intercepts of 3x – 4y – 12 = 0. Then graph the equation.

3x – 4y – 12 = 0

+ 12 +12

• x-intercept – (x, 0); y-intercept – (0, y)

Example 3 Find the x and y intercepts of 3x – 4y – 12 = 0. Then graph the equation.

3x – 4y – 12 = 0

+ 12 +12

3x – 4y = 12

• x-intercept – (x, 0); y-intercept – (0, y)

Example 3 Find the x and y intercepts of 3x – 4y – 12 = 0. Then graph the equation.

3x – 4y – 12 = 0

+ 12 +12

3x – 4y = 12

x-int.

• x-intercept – (x, 0); y-intercept – (0, y)

Example 3 Find the x and y intercepts of 3x – 4y – 12 = 0. Then graph the equation.

3x – 4y – 12 = 0

+ 12 +12

3x – 4y = 12

x-int. 3x – 4(0) = 12

• x-intercept – (x, 0); y-intercept – (0, y)

Example 3 Find the x and y intercepts of 3x – 4y – 12 = 0. Then graph the equation.

3x – 4y – 12 = 0

+ 12 +12

3x – 4y = 12

x-int. 3x – 4(0) = 12

(y=0) 3x = 12

• x-intercept – (x, 0); y-intercept – (0, y)

Example 3 Find the x and y intercepts of 3x – 4y – 12 = 0. Then graph the equation.

3x – 4y – 12 = 0

+ 12 +12

3x – 4y = 12

x-int. 3x – 4(0) = 12

(y=0) 3x = 12

3 3

• x-intercept – (x, 0); y-intercept – (0, y)Example 3 Find the x and y intercepts of 3x – 4y –

12 = 0. Then graph the equation.3x – 4y – 12 = 0

+ 12 +12 3x – 4y = 12

x-int. 3x – 4(0) = 12 (y=0) 3x = 12

3 3 x = 4

• x-intercept – (x, 0); y-intercept – (0, y)Example 3 Find the x and y intercepts of 3x – 4y –

12 = 0. Then graph the equation.3x – 4y – 12 = 0

+ 12 +12 3x – 4y = 12

x-int. 3x – 4(0) = 12 (y=0) 3x = 12

3 3 x = 4 (4, 0)

• x-intercept – (x, 0); y-intercept – (0, y)Example 3 Find the x and y intercepts of 3x – 4y –

12 = 0. Then graph the equation.3x – 4y – 12 = 0

+ 12 +12 3x – 4y = 12

x-int. 3x – 4(0) = 12 (y=0) 3x = 12

3 3 x = 4 (4, 0)

• x-intercept – (x, 0); y-intercept – (0, y)Example 3 Find the x and y intercepts of 3x – 4y –

12 = 0. Then graph the equation.3x – 4y – 12 = 0

+ 12 +12 3x – 4y = 12

x-int. 3x – 4(0) = 12 yint. (y=0) 3x = 12

3 3 x = 4 (4, 0)

• x-intercept – (x, 0); y-intercept – (0, y)Example 3 Find the x and y intercepts of 3x – 4y –

12 = 0. Then graph the equation.3x – 4y – 12 = 0

+ 12 +12 3x – 4y = 12

x-int. 3x – 4(0) = 12 yint. 3(0) – 4y = 12(y=0) 3x = 12

3 3 x = 4 (4, 0)

• x-intercept – (x, 0); y-intercept – (0, y)Example 3 Find the x and y intercepts of 3x – 4y –

12 = 0. Then graph the equation.3x – 4y – 12 = 0

+ 12 +12 3x – 4y = 12

x-int. 3x – 4(0) = 12 yint. 3(0) – 4y = 12(y=0) 3x = 12 (x=0) -4y = 12

3 3 x = 4 (4, 0)

• x-intercept – (x, 0); y-intercept – (0, y)Example 3 Find the x and y intercepts of 3x – 4y –

12 = 0. Then graph the equation.3x – 4y – 12 = 0

+ 12 +12 3x – 4y = 12

x-int. 3x – 4(0) = 12 yint. 3(0) – 4y = 12(y=0) 3x = 12 (x=0) -4y = 12

3 3 -4 -4 x = 4 (4, 0)

• x-intercept – (x, 0); y-intercept – (0, y)Example 3 Find the x and y intercepts of 3x – 4y –

12 = 0. Then graph the equation.3x – 4y – 12 = 0

+ 12 +12 3x – 4y = 12

x-int. 3x – 4(0) = 12 yint. 3(0) – 4y = 12(y=0) 3x = 12 (x=0) -4y = 12

3 3 -4 -4 x = 4 y = -3 (4, 0)

• x-intercept – (x, 0); y-intercept – (0, y)Example 3 Find the x and y intercepts of 3x – 4y –

12 = 0. Then graph the equation.3x – 4y – 12 = 0

+ 12 +12 3x – 4y = 12

x-int. 3x – 4(0) = 12 yint. 3(0) – 4y = 12(y=0) 3x = 12 (x=0) -4y = 12

3 3 -4 -4 x = 4 y = -3 (4, 0) (0, -

3)

• x-intercept – (x, 0); y-intercept – (0, y)Example 3 Find the x and y intercepts of 3x – 4y –

12 = 0. Then graph the equation.3x – 4y – 12 = 0

+ 12 +12 3x – 4y = 12

x-int. 3x – 4(0) = 12 yint. 3(0) – 4y = 12(y=0) 3x = 12 (x=0) -4y = 12

3 3 -4 -4 x = 4 y = -3 (4, 0) (0, -

3)

(4,0) (0,-3)x-int. = y-int. =

(4,0) (0,-3)x-int. = y-int. =

(4,0) (0,-3)x-int. = y-int. =