Download - 2 expressions and linear expressions

Expressions

Example B.a. We order pizzas from Pizza Grande. Each pizza is $3. How much would it cost for 4 pizzas? For x pizzas?

Expressions

Example B.a. We order pizzas from Pizza Grande. Each pizza is $3. How much would it cost for 4 pizzas? For x pizzas?For 4 pizzas, it would cost 3 * 4 = $12,

Expressions

Example B.a. We order pizzas from Pizza Grande. Each pizza is $3. How much would it cost for 4 pizzas? For x pizzas?For 4 pizzas, it would cost 3 * 4 = $12, for x pizzas it would cost 3 * x = $3x.

Expressions


b. There is $10 delivery charge. How much would it cost us in total if we want the x pizzas delivered?

Expressions


b. There is $10 delivery charge. How much would it cost us in total if we want the x pizzas delivered?In total, it would be 3x + 10 in $..

Expressions



Expressions

Formulas such as “3x” or “3x + 10” are called expressions in mathematics.



Expressions

Formulas such as “3x” or “3x + 10” are called expressions in mathematics. Mathematical expressions are calculation procedures and they are written with numbers, variables, and operation symbols.



Expressions

Formulas such as “3x” or “3x + 10” are called expressions in mathematics. Mathematical expressions are calculation procedures and they are written with numbers, variables, and operation symbols. Expressions calculate outcomes.



Expressions


The simplest type of expressions are of the form ax + b where a and b are numbers.



Expressions


The simplest type of expressions are of the form ax + b where a and b are numbers. These are called linear expressions.


b. There is $10 delivery charge. How much would it cost us in total if we want the x pizzas delivered?In total, it would be 3x + 10 in $.

Expressions


The simplest type of expressions are of the form ax + b where a and b are numbers. These are called linear expressions. The expressions “3x” or “3x + 10” are linear,


b. There is $10 delivery charge. How much would it cost us in total if we want the x pizzas delivered?In total, it would be 3x + 10 in $.

Expressions


The simplest type of expressions are of the form ax + b where a and b are numbers. These are called linear expressions. The expressions “3x” or “3x + 10” are linear, the expressions “x2 + 1” or “1/x” are not linear.

Expressions Combining Linear Expressions

Expressions Combining Linear ExpressionsEvery expression is the sum of simpler expressions and each of these addend(s) is called a term.


There are two terms in the linear expression ax + b.


There are two terms in the linear expression ax + b.

There are three terms in the expression ax2 + bx + c

Expressions Combining Linear ExpressionsEvery expression is the sum of simpler expressions and each of these addend(s) is called a term. Each term is named by its variable-component.There are two terms in the linear expression ax + b.



the x-term the constant term





the x2-term




the x2-term the constant termthe x-term

Just as 3 apples + 5 apples = 8 apples we may combine 3x + 5x = 8x, –3x – 5x = –8x,





Just as 3 apples + 5 apples = 8 apples we may combine 3x + 5x = 8x, –3x – 5x = –8x, 3x – 5x = –2x, and –3x + 5x = 2x.





Just as 3 apples + 5 apples = 8 apples we may combine 3x + 5x = 8x, –3x – 5x = –8x, 3x – 5x = –2x, and –3x + 5x = 2x. The x-terms can't be combined with the number terms because they are different type of items just as 2 apple + 3 banana = 2 apple + 3 banana (or 2A + 3B),i.e. the expression can’t be condensed further.





Just as 3 apples + 5 apples = 8 apples we may combine 3x + 5x = 8x, –3x – 5x = –8x, 3x – 5x = –2x, and –3x + 5x = 2x. The x-terms can't be combined with the number terms because they are different type of items just as 2 apple + 3 banana = 2 apple + 3 banana (or 2A + 3B),i.e. the expression can’t be condensed further.Hence the expression “2 + 3x” stays as “2 + 3x”, it's not “5x”.





ExpressionsExample C. Combine.

2x – 4 + 9 – 5x


2x – 4 + 9 – 5x= 2x – 5x – 4 + 9


2x – 4 + 9 – 5x= 2x – 5x – 4 + 9= –3x + 5

For the x-term ax, the number “a” is called the coefficient of the term.


2x – 4 + 9 – 5x= 2x – 5x – 4 + 9= –3x + 5

For the x-term ax, the number “a” is called the coefficient of the term. When we multiply a number with an x-term, we multiply it with the coefficient.


2x – 4 + 9 – 5x= 2x – 5x – 4 + 9= –3x + 5

For the x-term ax, the number “a” is called the coefficient of the term. When we multiply a number with an x-term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x,


2x – 4 + 9 – 5x= 2x – 5x – 4 + 9= –3x + 5

For the x-term ax, the number “a” is called the coefficient of the term. When we multiply a number with an x-term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x.


2x – 4 + 9 – 5x= 2x – 5x – 4 + 9= –3x + 5

For the x-term ax, the number “a” is called the coefficient of the term. When we multiply a number with an x-term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x. We may multiply a number with an expression and expand the result by the distributive law.


2x – 4 + 9 – 5x= 2x – 5x – 4 + 9= –3x + 5

For the x-term ax, the number “a” is called the coefficient of the term. When we multiply a number with an x-term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x. We may multiply a number with an expression and expand the result by the distributive law. Distributive LawA(B ± C) = AB ± AC = (B ± C)A


2x – 4 + 9 – 5x= 2x – 5x – 4 + 9= –3x + 5



2x – 4 + 9 – 5x= 2x – 5x – 4 + 9= –3x + 5

Example D. Expand then simplify.

a. –5(2x – 4)



2x – 4 + 9 – 5x= 2x – 5x – 4 + 9= –3x + 5


a. –5(2x – 4) = –5(2x) – (–5)(4)



2x – 4 + 9 – 5x= 2x – 5x – 4 + 9= –3x + 5


a. –5(2x – 4) = –5(2x) – (–5)(4) = –10x + 20

b. 3(2x – 4) + 2(4 – 5x)Expressions

b. 3(2x – 4) + 2(4 – 5x) expand

Expressions

b. 3(2x – 4) + 2(4 – 5x) expand = 6x

Expressions

b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12

Expressions

b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12 + 8

Expressions

b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12 + 8 – 10x

Expressions

b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12 + 8 – 10x = –4x – 4

Expressions

b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12 + 8 – 10x = –4x – 4

Expressions

Distributive law gives us the option of expanding the content of the parentheses i.e. extract items out of containers.

b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12 + 8 – 10x = –4x – 4

Expressions


Example E. A store sells two types of gift boxes Regular and Deluxe. The Regular has 12 apples and 8 bananas, the Deluxe has 24 apples and 24 bananas. We have 3 boxes of Regular and 4 boxes of Deluxe. How many apples and bananas are there?

b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12 + 8 – 10x = –4x – 4

Expressions


Let A stands for apple and B stands for banana,


b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12 + 8 – 10x = –4x – 4

Expressions


Let A stands for apple and B stands for banana, then Regular = (12A + 8B) and Deluxe = (24A + 24B).


b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12 + 8 – 10x = –4x – 4

Expressions


Let A stands for apple and B stands for banana, then Regular = (12A + 8B) and Deluxe = (24A + 24B). Three boxes of Regular and four boxes of Deluxe is 3(12A + 8B) + 4(24A + 24B)


b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12 + 8 – 10x = –4x – 4

Expressions


Let A stands for apple and B stands for banana, then Regular = (12A + 8B) and Deluxe = (24A + 24B). Three boxes of Regular and four boxes of Deluxe is 3(12A + 8B) + 4(24A + 24B) = 36A + 24B + 96A + 96B = 132A + 120B


b. 3(2x – 4) + 2(4 – 5x) expand = 6x – 12 + 8 – 10x = –4x – 4

Expressions


Let A stands for apple and B stands for banana, then Regular = (12A + 8B) and Deluxe = (24A + 24B). Three boxes of Regular and four boxes of Deluxe is 3(12A + 8B) + 4(24A + 24B) = 36A + 24B + 96A + 96B = 132A + 120B Hence we have 132 apples and 120 bananas.


ExpressionsWe usually start with the innermost set of parentheses to simplify an expression that has multiple layers of parentheses.

ExpressionsWe usually start with the innermost set of parentheses to simplify an expression that has multiple layers of parentheses. In mathematics, ( )’s, [ ]’s, and { }’s are often used to distinguish the layers.

ExpressionsWe usually start with the innermost set of parentheses to simplify an expression that has multiple layers of parentheses. In mathematics, ( )’s, [ ]’s, and { }’s are often used to distinguish the layers. This can't be the case for calculators or software where [ ] and { } may have other meanings.

ExpressionsWe usually start with the innermost set of parentheses to simplify an expression that has multiple layers of parentheses. In mathematics, ( )’s, [ ]’s, and { }’s are often used to distinguish the layers. This can't be the case for calculators or software where [ ] and { } may have other meanings. Always simplify the content of a set of parentheses first before expanding it.


Example F. Expand.

–3{–3x – [5 – 2(– 4x – 6)] – 4}


Example F. Expand.

–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,


= –3{–3x – [5 + 8x + 12] – 4}

Example F. Expand.

–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,


= –3{–3x – [5 + 8x + 12] – 4}

Example F. Expand.

–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,

simplify,


= –3{–3x – [5 + 8x + 12] – 4}

= –3{–3x – [17 + 8x] – 4}

Example F. Expand.

–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,

simplify,


= –3{–3x – [5 + 8x + 12] – 4}

= –3{–3x – [17 + 8x] – 4}

Example F. Expand.

–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,

simplify,

expand,


= –3{–3x – [5 + 8x + 12] – 4}

= –3{–3x – [17 + 8x] – 4}

Example F. Expand.

= –3{– 3x – 17 – 8x – 4}

–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,

simplify,

expand,


= –3{–3x – [5 + 8x + 12] – 4}

= –3{–3x – [17 + 8x] – 4}

Example F. Expand.

= –3{– 3x – 17 – 8x – 4}

–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,

simplify,

expand,

simplify,


= –3{–3x – [5 + 8x + 12] – 4}

= –3{–3x – [17 + 8x] – 4}

Example F. Expand.

= –3{– 3x – 17 – 8x – 4}

–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,

simplify,

expand,

simplify,

= –3{–11x – 21}


= –3{–3x – [5 + 8x + 12] – 4}

= –3{–3x – [17 + 8x] – 4}

Example F. Expand.

= –3{– 3x – 17 – 8x – 4}

–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,

simplify,

expand,

simplify,

= –3{–11x – 21} expand,

= 33x + 63

ExpressionsX-terms with fractional coefficients may be written in two ways,pq x or

pxq


pxq

( but not as ) pqx


pxq

( but not as ) since pqx

pq x =

pq

x1 = px

q*


pxq


pq x =

pq

x1 = px

q

Hence is the same as . 2x3

23 x

*


pxq


pq x =

pq

x1 = px

q

Example G. Evaluate if x = 6. 43

x


23 x

*


pxq


pq x =

pq

x1 = px

q


x

we get Set x = (6) in 43


23 x

*

43

x, (6)


pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6)


pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6) = 8


pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6) = 8

We may use the multiplier method to combine fraction terms i.e. multiply the problem by the LCD and divide by the LCD.


pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6) = 8


Example H. Combine 43

x + 54

x


pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6) = 8


43 x + 5

4x


x + 54

x

Multiply and divide by their LCD =12,


pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6) = 8


= ( )12 / 12

43 x + 5

4x

43 x + 5

4x


x + 54

x



pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6) = 8


= ( )12 / 12

43 x + 5

4x

43 x + 5

4x expand and cancel the denominators,


x + 54

x



pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6) = 8


4= ( )12 / 12

43 x + 5

4x

43 x + 5



x + 54

x



pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6) = 8


4 3= ( )12 / 12

43 x + 5

4x

43 x + 5



x + 54

x



pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6) = 8


4 3= ( )12 / 12

43 x + 5

4x

43 x + 5


= (4*4 x + 5*3x) / 12


x + 54

x



pxq


pq x =

pq

x1 = px

q


x


2


23 x

*

43

x, (6) = 8


31x12

4 3= ( )12 / 12

43 x + 5

4x

43 x + 5


= (4*4 x + 5*3x) / 12 = (16x + 15x) /12 =


x + 54

x


Exercise A. Combine like terms and simplify the expressions.Expressions

1. 3x + 5x 2. 3x – 5x 3. –3x – 5x 4. –3x + 5x

5. 3x + 5x + 4 6. 3x – 5 + 2x 7. 8 – 3x – 5

8. 8 – 3x – 5 – x 9. 8x – 4x – 5 – 2x 10. 6 – 4x – 5x – 2

11. 3A + 4B – 5A + 2B 12. –8B + 4A – 9A – B

B. Expand then simplify the expressions.

13. 3(x + 5) 14. –3(x – 5) 15. –4(–3x – 5) 16. –3(6 + 5x)

25. 3(A + 4B) – 5(A + 2B) 26. –8(B + 4A) + 9(2A – B)

17. 3(x + 5) + 3(x – 5) 18. 3(x + 5) – 4(–3x – 5)

19. –9(x – 6) + 4(–3 + 5x) 20. –12(4x + 5) – 4(–7 – 5x)

21. 7(8 – 6x) + 4(–3x + 5) 22. 2(–14x + 5) – 4(–7x – 5)

23. –7(–8 – 6x) – 4(–3 – 5x) 24. –2(–14x – 5) – 6(–9x – 2)

27. 11(A – 4B) – 2(A – 12B) 28. –6(B – 7A) – 8(A – 4B)

ExpressionsC. Starting from the innermost ( ) expand and simplify.

29. x + 2[6 + 4(–3 + 5x)] 30. –5[ x – 4(–7 – 5x)] + 6

31. 8 – 2[4(–3x + 5) + 6x] + x 32. –14x + 5[x – 4(–5x + 15)]

33. –7x + 3{8 – [6(x – 2) –3] – 5x}

34. –3{8 – [6(x – 2) –3] – 5x} – 5[x – 3(–5x + 4)]

35. 4[5(3 – 2x) – 6x] – 3{x – 2[x – 3(–5x + 4)]}

23 x + 3

4x36.

43 x – 3

4x37.

38 x – 5

6x39.5

8x + 1

6x38. – –

D. Combine using the LCD-multiplication method

40. Do 36 – 39 by the cross–multiplication method.

Expressions

42. As in example D with gift boxes Regular and Deluxe, the Regular contains12 apples and 8 bananas, the Deluxe has 24 apples and 24 bananas. For large orders we may ship them in crates or freight-containers where a crate contains 100 boxes Regular and 80 boxes Deluxe and a container holds 150 Regular boxes and 100 Deluxe boxes. King Kong ordered 4 crates and 5 containers, how many of each type of fruit does King Kong have?

41. As in example D with gift boxes Regular and Deluxe, the Regular contains12 apples and 8 bananas, the Deluxe has 24 apples and 24 bananas. Joe has 6 Regular boxes and 8 Deluxe boxes. How many of each type of fruit does he have?

Download - 2 expressions and linear expressions

Top Related