Download 2.2 – Linear Equations

2.2 – Linear Equations
2.2 – Linear Equations
Linear equation
2.2 – Linear Equations
Linear equation – equation with only addition,
2.2 – Linear Equations
Linear equation – equation with only addition,
subtraction,
2.2 – Linear Equations
Linear equation – equation with only addition,
subtraction, multiplication,
2.2 – Linear Equations
Linear equation – equation with only addition,
subtraction, multiplication,
and division of a variable by
a number.
2.2 – Linear Equations
Linear equation – equation with only addition,
subtraction, multiplication,
and division of a variable by
a number.
2.2 – Linear Equations
Linear equation – equation with only addition,
subtraction, multiplication,
and division of a variable by
a number.
2.2 – Linear Equations
Linear equation – equation with only addition,
subtraction, multiplication,
and division of a variable by
a number.
Linear Eqs.
2.2 – Linear Equations
Linear equation – equation with only addition,
subtraction, multiplication,
and division of a variable by
a number.
Linear Eqs.
5x – 3y = 7
2.2 – Linear Equations
Linear 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 Equations
Linear 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 Equations
Linear 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 Equations
Linear 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 Equations
Linear 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 = -8
x=9
6s = -3t – 15
y = ½x
2.2 – Linear Equations
Linear 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 = -8
x=9
6s = -3t – 15
y = ½x
2.2 – Linear Equations
Linear 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 = -8
x=9
y = √x + 5
6s = -3t – 15
y = ½x
2.2 – Linear Equations
Linear 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 = -8
x=9
y = √x + 5
6s = -3t – 15
y = ½x
2.2 – Linear Equations
Linear 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 = -8
x=9
y = √x + 5
6s = -3t – 15
x + xy = 1
y = ½x
2.2 – Linear Equations
Linear 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 = -8
x=9
y = √x + 5
6s = -3t – 15
x + xy = 1
y = ½x
2.2 – Linear Equations
Linear 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 = -8
x=9
y = √x + 5
6s = -3t – 15
x + xy = 1
y = ½x
y=1
x
2.2 – Linear Equations
Linear 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 = -8
x=9
y = √x + 5
6s = -3t – 15
x + xy = 1
y = 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
(b) g(x) = x4 – 5
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
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
+2x +2x
2x + y = 3
A=2,B=1,&C=3
(b) ⅜x = 3y + 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,&C=3
(b) ⅜x = 3y + 2
-3y -3y
• 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
(b) ⅜x = 3y + 2
-3y -3y
⅜x – 3y = 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,&C=3
(b) ⅜x = 3y + 2
-3y -3y
⅜x – 3y = 2
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
+2x +2x
2x + y = 3
A=2,B=1,&C=3
(b) ⅜x = 3y + 2
-3y -3y
⅜x – 3y = 2
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
+2x +2x
2x + y = 3
A=2,B=1,&C=3
(b) ⅜x = 3y + 2
-3y -3y
⅜x – 3y = 2
8(⅜x – 3y) = (2)8
3x – 24y = 16
A=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,B=1,&C=3
(b) ⅜x = 3y + 2
-3y -3y
⅜x – 3y = 2
8(⅜x – 3y) = (2)8
3x – 24y = 16
A=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
+2x +2x
2x + y = 3
A=2,B=1,&C=3
(b) ⅜x = 3y + 2
-3y -3y
⅜x – 3y = 2
8(⅜x – 3y) = (2)8
3x – 24y = 16
A=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)
x-int. = (4,0)
y-int. = (0,-3)
x-int. = (4,0)
y-int. = (0,-3)
x-int. = (4,0)
y-int. = (0,-3)