Algebra 1 Notes SOL A.4 Equations Mrs. Grieser - Loudoun County Public ...

Algebra 1 Notes SOL A.4 Equations

Mrs. Grieser

Name: __________________________________________ Date: _______________ Block: _______

Equations An equation is an open sentence where two expressions are set _____________.

Equations may have one or more unknowns (_______________) which we may try to solve.

Solving an equation means finding the value(s) of variable(s) that make the equation ____________.

Equations that have the same solution(s) are called ____________________.

We use ______________________ operations to solve equations.

We justify the steps in solving equations by using field properties.

Solving Equations "Isolate" the variable, justifying steps using field properties (properties of equality):

1) Put variables on one side of the = and numbers on the other by isolating x:

perform inverse operations (add, subtract, multiply, or divide)

2) Whatever you do to one side of the equation, you do to the other

3) Verify solutions

substitute solution in original equation (DO NOT SKIP THIS STEP!!!!!!) S

Field Properties of Equality:

Property of Equality Reflexive Property of Equality

Algebra (for real numbers a, b, c) a = a

Example

Symmetric Property of Equality If a = b, then b = a

Transitive Property of Equality If a = b and b = c, then a = c

Substitution Property of Equality If a = b, then a can be substituted for b

Addition Property of Equality

If a = b, then a + c = b + c

Subtraction Property of Equality If a = b, then a ? c = b - c

Multiplication Property of Equality

Division Property of Equality

If a = b, then ac = bc If a = b and c 0, then a b

cc

Algebra 1 Notes SOL A.4 Equations

Solve the equations:

a) x ? 4 = 6

x ? 4 + 4 = 6 + 4 x = 10 VERIFY: Is 10 a solution? (10) ? 4 = 6

6=6

b) x + 3 = -5

x + 3 ? 3 = -5 ? 3 x = -8 VERIFY: Is -8 a solution? (-8) + 3 = -5

-5 = -5

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c) 4x = 16 4x ? 4 = 16 ? 4 x = 4 VERIFY: Is 4 a solution? 4(4) = 16 16 = 16

Solve the equations, justifying steps:

a) Solve x + 4 = 10 1) x + 4 = 10 2) x + 4 ? 4 = 10 ? 4

3) x = 6

Given Subtraction Property of Equality Simplify

b) Solve 2 x = 4 3

1) 2 x = 4 3

Given

2) 3 2 x 3 4 Mult. Property of Equality 23 2

3) x = 6

Simplify

Two-Step Equations

SAME! Put variables on one side of the = and numbers on the other:

How to Solve Two-Step Equations 1) Clear parentheses (distribute) and combine like terms if necessary 2) Do add/subtract steps first

3) Do multiply/divide steps

Examples:

a) 5x + 9 = 24

5x + 9 ? 9 = 24 ? 9 __________________

5x = 15

__________________

5x 15 55

__________________

x = 3

__________________

VERIFY:

b) x 5 11 2

x 5 5 11 5 ________________ 2

x 16 2

________________

2 x 216 2

________________

x = 32

_________________

VERIFY:

c) 3x + 2x = 15

d) 4(x - 6) = 32

VERIFY:

VERIFY:

Algebra 1 Notes SOL A.4 Equations

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Practice - solve the equations, justifying steps with field properties:

a) x + 9 = 25

b) -4x = -20

c) 3 x 3 5

d) 2x + 3 = -9

e) a 4 6 3

f) -1 = z 3 37

g) 7x ? 4x = 21

h) 3(x + 2) = 9

Multistep Equations and Equations with Variables on Both Sides

How to Solve Multi-Step Equations and Equations with Variables on Both Sides 1) Clear parentheses (distribute) and combine like terms if necessary 2) Add/subtract variable terms so that variable is on one side (NEW STEP!) 3) Do add/subtract steps first 4) Do multiply/divide steps

Example: Solve 7 ? 3x = 4x ? 7

1) 7 ? 3x = 4x ? 7 2) 7 ? 3x - 4x = 4x ? 7 - 4x 3) 7 ? 7x = - 7 4) 7 -7x - 7 = -7 - 7

Given ________________________________________________ ________________________________________________ ________________________________________________

5) -7x = -14 6) - 7x -14

7 -7 7) x = 2

________________________________________________ ________________________________________________

________________________________________________

Examples: Solve the equation, justifying steps...

a) 9x ? 5 = 1 (16x + 60) 4

b) 3(x + 12) ? x = 4(2 ? x) ? 3x + 2x

9x ? 5 = 4x + 15 ______________________

5x ? 5 = 15

______________________

5x = 20

______________________

x = 4

_____________________

Algebra 1 Notes SOL A.4 Equations

Special cases:

Identity case (solutions are all real numbers):

2x + 6 = 2(x + 3)

2x + 6 = 2x + 6 Distribute

0 = 0

Subtraction Prop. of =

Where did the variable go??

When we get a TRUE statement at the end when the variable "disappears," EVERY x is a solution (the two sides of the equation are identical!).

ALL REAL NUMBERS are solutions.

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No solution case:

3x = 3(x + 4)

3x = 3x + 12 Distribute

0 = 12

Subtraction Prop. of =

Where did the variable go??

When we get a FALSE statement at the end when the variable "disappears," there are NO SOLUTIONS. There is no value of x that makes the equation true.

Other Equations: Solve: a) x 3

42

b) x 2 x 1 3

Cross Product Property: If a c then ad = bc bd

REMEMBER TO GROUP NUMERATORS AND DENOMINATORS (use parentheses)

You Try: Solve the equation if possible; if not, write "all real numbers" or "no solution"...

a) 8x + 5 = 6x + 1

b) x + 1 = 3x - 1

c) 9x = 6(x + 4)

d) 7(x + 7) = 5x + 59

e) 2 ? 15x = 5(-3x + 2)

f) 12y + 6 = 6(2y + 1)

g) 5(x + 2) = 3 (5 + 10x) 5

h) 40 + 14x = 2(-4x ? 13)

i) 2(3x + 2) = 1 (12x + 8) 2

j) x 9 23

k) 4 8 x-8 2

l) 2 x 5 21- x

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