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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. =