solving equations

Solving Equations Solving Equations

1) open sentence2) equation3) solution

Translate verbal expressions into algebraic expression and equations and vice versa. Solve equations using the properties of equality.

Solving Equations Solving Equations

A mathematical sentence (expression) containing one or more variables is called an open sentence.

A mathematical sentence stating that two mathematical expressions are equalis called an _________.equation

Open sentences are neither true nor false until the variables have been replaced by numbers.

Each replacement that results in a true statement is called a ________ of theopen sentence.

solution

To solve equations, we can use properties of equality.Some of these equivalence relations are listed in the following table.

Properties of Equality Property Symbol Example

Reflexive

Symmetric

Transitive

Substitution

For any real number a,

a = a,

For all real numbers a and b,

If a = b, then

For all real numbers a, b, and c.

If a = b, and b = c, then

If a = b, then a may be replacedby b and b may be replaced by a.

b = a

a = c

– 5 + y = – 5 + y

If 3 = 5x – 6, then 5x – 6 = 3

If 2x + 1 = 7 and 7 = 5x – 8

then, 2x + 1 = 5x – 8

If (4 + 5)m = 18

then 9m = 18

Solving Equations Solving Equations

Sometimes an equation can be solved by adding the same number to each side or bysubtracting the same number from each side or by multiplying or dividing each side by the same number.

Addition and Subtraction Properties of Equality

For any real numbers a, b, and c, if a = b, then

a = b+ c + c a = b- c - c

Example:

If x – 4 = 5, then x – 4 = 5+ 4 + 4

If n + 3 = –11, then n + 3 = –11– 3 – 3

Solving Equations Solving Equations

Sometimes an equation can be solved by adding the same number to each side or bysubtracting the same number from each side or by multiplying or dividing each side by the same number.

Multiplication and Division Properties of Equality

For any real numbers a, b, and c, if a = b, then

a = b· c · c a = b

Example:

4 4

c c

then64m

If 6 4m then6y3 If - 6 3 y

-3 -3

Solving Equations Solving Equations

• What will we discuss?What are the parts of an equationWhat does it mean to solve an equationHow do we use inverse operations to solve

equationsHow to solve simple and complex equations

Chapter 5: Solving Equations

What Does it Mean to Solve an What Does it Mean to Solve an Equation?Equation?

To solve an equation means to find every number that makes the equation true.

We do this by adding or subtracting to each side of the equation … but always keep it balanced!

What are the parts of an equation?What are the parts of an equation?

Let’s first take a look at an Let’s first take a look at an equation and identify its partsequation and identify its parts

3 1 2 3 6x x

Coefficient

ConstantVariable

So do we just use trial and error to So do we just use trial and error to find the right value?find the right value?

No. No. We can use We can use inverse operationsinverse operations to to

isolate, or solve for, the variable’s isolate, or solve for, the variable’s value.value.

Inverse operations? Think about it …Inverse operations? Think about it …The inverse operation of addition is The inverse operation of addition is subtractionsubtraction. And the inverse . And the inverse operation of multiplication is operation of multiplication is divisiondivision..

Solving 1 Step EquationsSolving 1 Step Equations

How much does the suitcase weigh

in terms of blocks?

B=Blocks S=Suitcase

Equation: 6B + S = 9B

-6B

-6B

3BS =What is the weight of the suitcase if each

block has a weight of 2lbs. ?

S = 3 (2) = 6 lbs.

So how do we solve equations with So how do we solve equations with inverse operations?inverse operations?

Let’s take a look at a simple equationLet’s take a look at a simple equation

Step 1:- 13

Answer:

2113 x- 13

8x

Now that we have solved the equation, let’s check the solution:

2121

2113 x

21138

So how do we solve equations with So how do we solve equations with inverse operations?inverse operations?

Let’s take a look at a simple equationLet’s take a look at a simple equation

Step 1:+ 5

Answer:

125 y+ 5

17y

Now that we have solved the equation, let’s check the solution:

1212

125 y

12517

So how do we solve equations with So how do we solve equations with inverse operations?inverse operations?

Let’s take a look at a simple equationLet’s take a look at a simple equation

Step 1:

Step 2:

25

Answer:

7525 W25

25

75W

3W

Now that we have solved the equation, let’s check the solution:

7525 W

75)3(25

7575

So how do we solve equations with So how do we solve equations with inverse operations?inverse operations?

Let’s take a look at a simple equationLet’s take a look at a simple equation

Step 1:

Step 2:

(16)

Answer:

416

A

(16)

)16(4A

64A

Now that we have solved the equation, let’s check the solution:

44

416

A

416

64

1 Step Equation1 Step Equation

X + 11 = 9 X - 37 = 52 3X = 72-11 -

11X = -2

3 3

X = 24

1

1

20 + h = 41 17 - s = 27

37 37

52 37 - X

89X

20-20-

41 h 20

21h1717

2717

s

10 sThis is the same as -

1S=10

101

10

1

s

1 Step Equations Continued…1 Step Equations Continued…6X = 42

25

P = 34

6 6

1X=7

or x = 7

1

1

47

x

1

74

1

7

7

x

28x

213

s

1

213

s1321 sCross Multiply

21

3s

7

1

Multiply by the

reciprocal of 2/5

2

5

4

3

2

5

5

2 p

8

151 p

Multi Step Equations

Solve:

8m – 10 = 36

423

31176w

8m – 10 = 36

8m = 468 8

m =

+ 10 + 10 1717

31176

w

416w

1

64

1

6 16

w

84w

Multi Step EquationsMulti Step Equations

5x 2 = x + 4 Solve:

5x 2 = x + 4

Notice that there are variables on both sides

5x = x + 6

Get rid of the -2 on the left side

Simplify

5x = x + 6Get rid of the x on the right side4x = 6Get rid of the coefficient of x

4 4

23x = Simplify

Simplify

+ 2 + 2

– x– x

Solving a ProportionSolving a Proportion

Solve the proportion belowSolve the proportion below

60

126

C)12()60(6 C

C12360 12 12

C30

Solving a ProportionSolving a Proportion

Solve the proportion belowSolve the proportion below

852

13 a )(52)8(13 a

a52104 52 52

a2

Checking the Solution to a Checking the Solution to a ProportionProportion

Let’s check the solution to the Let’s check the solution to the proportion we solved on the last slideproportion we solved on the last slide

852

13 a

852

13 2

4

1

4

1

Using Proportions to Solve Using Proportions to Solve ProblemsProblems

You get 46 miles to a gallon of gas. You get 46 miles to a gallon of gas. How far can you go on 16 gallons of How far can you go on 16 gallons of gas?gas?

m

16

46

1 )16(461 m

736m

Multi-step SolutionsMulti-step Solutions

Let’s take a look at our original equationLet’s take a look at our original equation

3 1 2 3 6x x

3 2 4x x

x 6

4 2 4x

Step 1:-12 -12

Step 2:+x +x

Step 3:4 4

Answer:

Multi-step SolutionsMulti-step Solutions(involving distribution)(involving distribution)

Consider the following equationConsider the following equation

Step 1:

Step 2:

Step 3:

Answer:

42)5(6 x

+3042306 x

+30

726 x6 6

12x

Finding Variations of FormulasFinding Variations of Formulas

Solve the formula for Solve the formula for rr..

trs 2

s trs 2

s s

tr 2

2s

tr

22