Grade 7 Math Unit 6: Equations

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Review of Expressions versus Equations

Algebraic Expression – is a mathematical sentence with numbers and/or variables connected by +,−,×,÷ . ( There is NO equal sign. )

Ex: 8m + 2 m is the variable, 8 is the numerical coefficient and 2 is the constant term

Equation - is a statement where two expressions are equal. An equation contains an equal sign, an expression does not.

4x = 8 and 3x – 1 = 71 are equations

4x and 3x – 1 are expressions

Questions

1. Which are expressions and which are equations? Explain A). 3n + 12 B). 3n = 12 C). 5x + 2 D). 5x + 2 = 27

Answer:

2. Write an equation for each sentence.

A). The sum of 10 and a number is 15.

B). The product of a number and nine is 63.

C). Eleven decreased by two times a number is 1.

Sum means Add

Product means Multiply

Decrease means Subtract

D). The quotient of a number and two, increased by five, is 3.

Sec 6.1: Solving Equations.

In this unit we will look at different ways of solving equations1. Systematic Trial2. Inspection3. Models (balance & algebra tiles)4. Algebra

Consider 3d+5=17

1). Systematic Trial► means choosing a value for the variable and evaluating. “Guess and check”

Try d = 2 3d + 5 = 17 Try d = 5 3d + 5 = 17 3× 2 + 5 = 17 3× 5 + 5 = 17 6 + 5 = 17 15 + 5 = 17 11 is not 17 20 is not 17 so d = 2 was too small so d = 5 was too big

Try d = 4 3d + 5 = 17 3× 4 + 5 = 17 12 + 5 = 17 17 = 17

Therefore d = 4 is the answer

2). Inspection ► means finding the value for the variable using addition, subtraction, multiplication and division.

3 d + 5 = 17 “Think it through”

What + 5 = 17

12 + 5 = 17

therefore 3d = 12 what times 3 equals 12 so d must equal 4.

Quotient means Divide Increased by means Add

Examples:

1. Use systematic trial to solve: 2d + 3 = 21

2. Use inspection to solve: 4m – 2 = 46 . Check your answer:

To complete on loose leaf: p.223 #1, 2, 4, 5, 6 and 10.

Grade 7 Math Equations

1. Ben has a large collection of baseball caps. Answer each question below.

a). If Ben takes y caps from a group of 18 caps, there are 12 caps left. How many caps did Ben take away? Write an equation and solve by inspection.

b). Ben put k caps in each of 6 piles. There are 108 caps altogether. How many caps did Ben put in each pile? Write an equation and solve by systematic trail.

c). Ben shares n caps equally among 9 piles. There are 6 caps in each pile. How many caps did Ben have? Write an equation and solve by inspection.

d). Ben combines p groups of 4 caps each into one large group. He then takes away 7 caps. There are 49 caps left. How many groups of 4 caps did Ben begin with? Write an equation and solve by systematic trail.

2. Perimeter Problems.

a). The perimeter of an equilateral triangle is 39 cm. Write and solve an equation to find the length of each side of the triangle.

b). The length of the side of a regular hexagon is 5 cm. Write and solve an equation to find the perimeter of the hexagon.

Sec 6.2: Using a Model to Solve Equations

Sometimes systematic trial and inspection are not the best ways to solve an equation. A balance can be used to model an equation.

* having both sides of the balance at the same height represents them being equal.

Write an equation to represent this balance.

20 = 10 + 5 + 5

Examples:

1. For each balance write an equation and find the unknown value. Verify your answer.

A).

Check:Answer: x + 3 + 5 = 7 + 10 9 + 3 + 5 = 7 + 10 x + 8 = 17 17 = 17 therefore x = 9 We’re Right !

B).

Check:Answer:

Make sure to always keep the balance equal. Each side must equal the same amount.Refer to the balances below and answer the questions.

20g 10g 5g 5g

? 3 5 7 10

8 11 ? 4

To verify an answer means to put your answer back into the problem and the balance should be equal.

1.

a). Is this balanced?

b). What if we added 3g to each side, is it still balanced?

** Therefore, we can add the same amount to each side of the balance and still keep it equal **2.

To find the value of x, we can remove 7 from the left side of the scale, However, we must keep it balanced, so we must remove 7 from the right side too!

Removing 7 from 25leaves 18 on the right side.

Algebra Workings:x + 7 = 25

x + 7 = 7 + 18 x + 7 – 7 = 7 + 18 – 7

x = 18

When you subtract from one side of the balance, you must subtract the same amount from the other side of the balance too! This keeps the balance equal.

More Balance Examples

12g 8g 4g

12g 3g 8g 4g 3g

x 7 25

x 7 7 18

x 18

To complete on loose leaf: p.229 #1a, b, c

Use algebra and balances to find the missing value.

Algebra Model

2w + 6 = 28

FINISH!!!!!

Examples:

w w 286

To complete on loose leaf: p.229 #1d

1. 3c = 15 c + c + c = 15 c + c + c = 5 + 5 + 5

c = 5

2. 2x + 1 = 7 x + x + 1 = 7 x + x + 1 = 6 + 1

x + x = 6 x + x = 3 + 3 x = 3

3. Area of a Rectangle Example

A = 32 m2 4m

?

A = b × h ( or length × width ) 32 = b × 4 32 = b + b + b + b 8 + 8 + 8 + 8 = b + b + b + b b = 8

splitting into equal groups is the same as dividing ...this can be done easier.

A = b × h

32 = b × 4

32 = b × 4 4 4

8 = b

Sec 6.3: Solving Equations Involving Integers

Remember from integersShaded = negative

Unshaded = positive

To complete on loose leaf: p.229 #2, 3, 4

−¿1 +1

Together this is a zero pair.

We represent the variable x, using the following tiles:

−¿x + x

We can also use Algebra tiles.They are different colors, but they mean the same thing.

+ 1 + xWe will always represent shadedas positive and unshaded as

−¿1 −¿ x negative when drawing tiles.

Examples:

1. Represent the equations using tiles.

a). x + 2 = 5 =

b). 2x + 1 = 3 =

c). 3x −¿ 1 = 8 =

* Remember that opposite integers, or opposite tiles, are ZERO.2. What is the opposite of each expression?

Shaded

= negative

Unshaded

= positive

a). - 3 +¿ 3

b).

+¿ 2 - 2

c).

- 2x +¿ 2x

3. Write the equation, then represent the equation using tiles.

Statement Equation Tilesone more than double

a number is seven

five more than a number is nine

triple a number decreased by four is

eight

twice a number increased by three is

five

six decreased by a number equals one

four less than double a number is six

Sec 6.3 and Sec 6.4: Using Algebra and Algebra Tiles to Solve Equations

To solve an equation means to find the correct number for the variable that makes the equation true.

For example: x + 2 = 5

By inspection, you know the answer is 3. Three makes this equation true.3 + 2 = 5

5 = 5

Using algebra or algebra tiles our goal is to get the variable, which is often x, by itself.When solving the equation we have to make sure the equation is always balanced.

Try: x + 3 = 7

x + 3 = 7

We need to get rid of + 3 so that x will be by itself. We can do this by making it zero.What can we add to +3 to make it zero? −¿ 3

x + 3 + (−¿ 3) = 7 + (−¿ 3)

If we add – 3 to one side ofthe equation, we must add itto the other side too! This keeps the equation balanced.

x + 3 + (−¿ 3) = 7 + (−¿ 3)

+ 4 Zero

Zero ZeroNow we are left with just x on the left side of the equation and +4 on the right side of the equation. Therefore,

x = + 4

Examples: Solve using algebra and algebra tiles.

Algebra Algebra Tiles

a). x + 1 = 3 =

x + 1 + (−¿ 1) = 3 + (−¿ 1) =

x + 1 + (−¿ 1) = 3 + (−¿ 1) =

x = 2 =

Check (verify) your answer.

x + 1 = 3 2 + 1 = 3 3 = 3

Algebra Algebra Tiles

b). x + 2 = −¿ 5 =

x + 2 + (−¿ 2) = −¿ 5 + (−¿ 2) =

x + 2 + (−¿ 2) = −¿ 5 + (−¿ 2) =

x = −¿ 7 =

Check (verify) your answer. x + 2 = −¿ 5 −¿ 7 + 2 = −¿ 5 −¿ 5 = −¿ 5

Algebra Algebra Tiles

c). x −¿ 1 = −¿ 2 =

x −¿ 1 + ( + 1) = −¿ 2 + (+¿ 1) =

x −¿ 1 + ( + 1) = −¿ 2 + (+¿ 1) =

Check! x −¿ 1 = −¿ 2 x = −¿ 1 −¿ 1 −¿ 1 =

−¿ 2 =

p. 234 #1, 2

Solving Equations in the form x + b = c

Complete the table by using algebra and algebra tiles to solve equations. Verify each answer to make sure it is correct.

A).Algebra Algebra Tiles Verify

x + 3 = 7

B).Algebra Algebra Tiles Verify

x + 2 = 5

C).Algebra Algebra Tiles

x - 1 = 3

D).Algebra Algebra Tiles Verify

x - 7 = 5

Solving Equations with more than 1x

a).Algebra Algebra Tiles

2x + 3 = 7

Verify the answer.

b).Algebra Algebra Tiles

3x = 6

Verify:

c).Algebra Algebra Tiles

2x + 1= 3

Verify:

d). Algebra Algebra Tiles

2x – 4 = 6

Verify:

Grade 7 Math Algebra

Topic: Solving Equations in the form ax = c and ax + b = c

1. Complete the table by using algebra and algebra tiles to solve equations. Verify each answer to make sure it is correct. A).

Algebra Algebra Tiles Verify

2x = 4

B).

Algebra Algebra Tiles Verify

3x = −¿ 6

C).

Algebra Algebra Tiles Verify

3x −¿ 2 = 7

D).

Algebra Algebra Tiles Verify

2x + 5 = 9

E). Algebra Algebra Tiles Verify

4x −¿ 3 = 9

F). Algebra Algebra Tiles Verify

2x −¿ 2 = 6

Word Problems

1. Three more than two times a number is 27.a). Write an equation. 2x + 3 = 27b). Solve the equation, using algebra. 2x + 3 = 27 2x + 3 + (−¿ 3) = 27 + (−¿ 3)

2x = 24 2x = 24

2 2 x = 12c). Verify. 2x + 3 = 27 2(12) + 3 = 27 24 + 3 = 27 27 = 27

2. My mother’s age is 4 more than two times my brother’s age. If my mom is 46 years old, how old is my brother?

a). Write an equation b). Solve. c). Verify

3. Eight years ago, Susan was 13 years old. How old is she now?

a). Write an equation b). Solve. c). Verify

4. Hannah borrowed money. She paid back $7, but still owes $5. How much did she borrow?

a). Write an equation b). Solve. c). Verify

To complete on loose leaf: p. 238 #1, 2, 3, 4, 5

Grade 7 Math Solving Equations

Use algebra to solve each equation. Show all necessary steps.

1). a + 2 = 6 2). b - 3 = 1 3). c - 2 = 5

4). d 5 = 5 5). e + 7 = 25 6). 4f = 8

7). 3g = 12 8). 5h + 5 = 10 9). 2j + 4 = 16

10). 6k 3 = 15 11). 3m + 12 = 18 12). 3n + 4 = 7

13). Show whether or not x = 7, is a solution to each equation.

a). 6x = 48

b). x = 1 7

c). 3x + 2 = 20

Solving Equations with Fractions

1. Laurie has 13 of a chocolate bar. It weighs 5g.

a). Write an equation.

13b=5 or

b3=5

b). Solve the equation using a balance, to find the weight of the whole bar.

bar

bar

Therefore, the whole bar weighs 15 g. b = 15 g

c). Verify. b = 5 15 = 5 3 3

2. Find p if p = 8. 2

Therefore, p = 16.

Verify p = 8 16 = 8 2 2

5 g

5 g5 g 5 g

8

8 8

3. Find x = 4 5

Therefore, x = 20 Verify 20 = 4 5

What do you notice from the following examples?

a). b = 5 b). p = 8 c). x = 4 d). x = 4 3 2 5 2 b = 15 p = 16 x = 20 x = 8

e). x = 7 f). x = 16 g). x = 10 3 4 5

x = 21 x = 64 x = 50

**** If you multiply the numbers you get the answer. This is called cross multiplying.

4

4 4 4 4 4

To complete on loose leaf: p.243 #2

To complete on loose leaf: p.243 #1abcd p. 238 #1, 2, 3, 4, 5

Fraction Word Problems

1. One half of Leah’s money paid for her movie ticket. This amount was $7. Write an equation, solve and verify, to find her original amount of money?

m = 7 2

m = 7 × 2 m = 14 Leah had $14 originally

2. Jordan ate one eighth of a pizza. This was 2 slices. How many slices of pizza was there? Equation Solve Verify

p = 2 8

p = 8 × 2 p = 16 There was 16 slices of pizza.

3. Mary baked some cookies and shared then equally among her 4 children. If each child got 3 cookies, how many did she bake? Equation Solve Verify

c = 3 4

c = 3 × 4 c = 12 Mary baked 12 cookies.

3. Brigitte is solving the equation f = 10. Here is her answer. 8 Answer: f = 10 8

f −¿ 8 = 10 −¿ 8 8

f = 2

a). Is her solution correct? Explain. ► No. If you verify her answer 2 ≠ 10. Instead of subtracting 8, she should of multiplied 10 × 8. 8

f = 10 f = 80 8

14 = 7 2

16 = 2 8

12 = 3 4

To complete on loose leaf: p.243 #3,4,6