Unit 3A Lesson 13-17 Classwork - Chenango Forks Middle … 3A Lesson 13-17...

Lesson 13-‐17: Expressions and Equations

Unit 8-‐3 CW Lessons 13-‐17

Expressions and Equations: Lessons 13-‐17

Name Date Classwork Book Math 8: Mr. Sanford

Mathematics 8: Mr. Sanford CW Lesson 3-‐13

Grade 8 Lesson 3-‐13: Writing Equations Using Symbols 3

Lesson 13: Writing Equations Using Symbols Example:

Write each of the following statements using symbolic language.

1. The sum of four consecutive even integers is −28.

2. A number is four times larger than the square of half the number.

3. Steven has some money. If he spends $9.00, then he will have !! of the amount he started with.

4. The sum of a number squared and three less than twice the number is 129.

5. Miriam read a book with an unknown number of pages. The first week, she read five less than !! of the pages. The second week,

she read 171 more pages and finished the book. Write an equation that represents the total number of pages in the book.

Lesson Summary

Begin all word problems by defining your variables. State clearly what you want each symbol to represent.

Written mathematical statements can be represented as more than one correct symbolic statement.

Break complicated problems into smaller parts or try working them with simpler numbers.


Grade 8 Lesson 3-‐13: Writing Equations Using Symbols 4

Problem Set Write each of the following statements using symbolic language.

1. Bruce bought two books. One book costs $4.00 more than three times the other. Together, the two books cost him $72.

2. Janet is three years older than her sister Julie. Janet’s brother is eight years younger than their sister Julie. The sum of all of their ages is 55 years.

3. The sum of three consecutive integers is 1,623.

4. One number is six more than another number. The sum of their squares is 90.

5. When you add 18 to !! of a number, you get the number itself.

6. When a fraction of 12 is taken away from 17, what remains exceeds one-‐third of seventeen by six.

7. The sum of two consecutive even integers divided by four is 189.5.


Grade 8 Lesson 3-‐14: Linear and Nonlinear Expressions in 𝑥 5

Lesson 14: Linear and Nonlinear Expressions in 𝒙

Example 1: Expressions

• A symbolic statement in 𝑥 with an equal sign is called . The equal sign divides the equation into two parts, the left side and the right side. The two sides are called .

• The chart below contains two types of expressions, and . • A expression cannot contain exponents other than 0 or 1.

Directions: Circle the linear expressions in the chart below.

Exercise:

Explain why 2𝑥! + 9𝑥 + 5 is not a linear expression in 𝑥.

Example 2: Terms

• A is a multiplication problem found in an expression. It is made up of a coefficient (number) and a variable often with an exponent.

• You can identify terms because they will be separated by addition or subtraction symbols. Consider the expression; 4 + 3𝑥!

a. How many terms are there and what are they?

b. How many terms are made up of just constants, what are they?

c. How many terms have coefficients, and what are they?

d. Is the expression linear or nonlinear? How do you know?



Example 3: More Terms

Consider each expression below. Identify how many terms are in each expression and whether the expression is linear or nonlinear.

a. 7𝑥 + 9 + 6 + 3𝑥

b. 5 + 9𝑥 ∙ 7 + 2𝑥!

c. 94 + 𝑥 + 4𝑥!! − 2

d. !!+ 9𝑥 − 4

Exercises

Write each of the following statements in Exercises 1–12 as a mathematical expression. State whether or not the expression is linear or nonlinear. If it is nonlinear, then explain why.

1. The sum of a number and four times the number.

2. The product of five and a number.

3. Twice a number subtracted from four times a number, added to 15.

4. The square of the sum of six and a number.

5. The cube of a positive number divided by the square of the same positive number.



Problem Set Directions: State whether the expression is linear or nonlinear. If it is nonlinear, then explain why.

1. 𝑛! − 3

2. 17 − !!

3. 13 + 2𝑏

4. 19𝑔 − !!!

5. 𝑡! − 𝑡 + 6

6. 3𝑓! + 7𝑓 − 3 − 3𝑓! − 2𝑓


Grade 8 Lesson 3-‐15: Linear Equations in 𝑥 8

Lesson 15: Linear Equations in 𝒙

Exercises

1. Is the equation a true statement when 𝑥 = −3; in other words, is −3 a solution to the equation 6𝑥 + 5 = 5𝑥 + 8 + 2𝑥? Explain.

2. Does 𝑥 = 12 satisfy the equation 16 − !!𝑥 = !

!𝑥 + 1? Explain.

3. Chad solved the equation 24𝑥 + 4 + 2𝑥 = 3(10𝑥 − 1) and is claiming that 𝑥 = 2 makes the equation true. Is Chad correct? Explain.



4. Lisa solved the equation 𝑥 + 6 = 8 + 7𝑥 and claimed that the solution is 𝑥 = − !!. Is she correct? Explain.

5. Angel transformed the following equation from 6𝑥 + 4 − 𝑥 = 2(𝑥 + 1) to 10 = 2(𝑥 + 1). He then stated that the solution to the equation is 𝑥 = 4. Is he correct? Explain.

Lesson Summary

Equations are statements about equality. If the expression on the left side of the equal sign has the same value as the expression on the right side of the equal sign, then you have a true equation.

A solution of a linear equation in 𝑥 is a number, such that when all instances of 𝑥 are replaced with the number, the left side will equal the right side. For example, 2 is a solution to 3𝑥 + 4 = 𝑥 + 8 because when 𝑥 = 2, the left side of the equation is

3𝑥 + 4 = 3(2) + 4 and the right side of the equation is

𝑥 + 8 = 2 + 8 Since 10 = 10, then 𝑥 = 2 is a solution to the linear equation 3𝑥 + 4 = 𝑥 + 8.



Problem Set 1. Given that 2𝑥 + 7 = 27 and 3𝑥 + 1 = 28, does 2𝑥 + 7 = 3𝑥 + 1? Explain.

2. Is −5 a solution to the equation 6𝑥 + 5 = 5𝑥 + 8 + 2𝑥? Explain.

3. Does 𝑥 = 1.6 satisfy the equation 6 − 4𝑥 = − !!? Explain.

4. Use the linear equation 3 𝑥 + 1 = 3𝑥 + 3 to answer parts (a)–(d).

a. Does 𝑥 = 5 satisfy the equation above? Explain.

b. Is 𝑥 = −8 a solution of the equation above? Explain.

c. Is 𝑥 = !! a solution of the equation above? Explain.


Grade 8 Lesson 3-‐16: Solving Linear Equations 11

Lesson 16: Solving Linear Equations Notes:

Example 1:

Solve the linear equation 2𝑥 − 6 = 4𝑥 for the number 𝑥. Check the validity of your solution.

Example 2:

Solve the linear equation !!+ 21 = 15. Check the validity of your solution.



Example 3:

Solve the linear equation 𝑥 + 13 + 𝑥 = 1 − 8𝑥 + 22. Check the validity of your solution.

Exercises For each problem, show your work, and check that your solution is correct. 1. Solve the linear equation 𝑥 + 𝑥 + 2 + 𝑥 + 4 + 𝑥 + 6 = −28. 2. Solve the linear equation 2 3𝑥 + 2 = 2𝑥 − 1 + 𝑥. State the property that justifies your first step and why you chose it.

3. Solve the linear equation 𝑥 − 9 = !!𝑥. State the property that justifies your first step and why you chose it.



Problem Set For each problem, show your work and check that your solution is correct. 1. Solve the linear equation 𝑥 + 4 + 3𝑥 = 72. State the property that justifies your first step and why you chose it. 2. Solve the linear equation 𝑥 + 3 + 𝑥 − 8 + 𝑥 = 55. State the property that justifies your first step and why you chose it. 3. Solve the linear equation 3𝑥 + 10 = 5𝑥 + 54. State the property that justifies your first step and why you chose it.

4. Solve the linear equation 12 − !!𝑥 = !

!𝑥. State the property that justifies your first step and why you chose it.


Grade 8 Lesson 3-‐17: Classification of Solutions 14

Lesson 17: Classification of Solutions Exercises Solve each of the following equations for 𝑥. 1. 7𝑥 − 3 = 5𝑥 + 5 2. 7𝑥 − 3 = 7𝑥 + 5 3. 7𝑥 − 3 = −3 + 7𝑥 4. 11𝑥 − 2𝑥 + 15 = 8 + 7 + 9𝑥



5. 3 𝑥 − 14 + 1 = −4𝑥 + 5 6. −3𝑥 + 32 − 7𝑥 = −2(5𝑥 + 10)

7. !!8𝑥 + 26 = 13 + 4𝑥

8. – 4𝑥 − 2 8𝑥 + 1 = − −2𝑥 − 10



Problem Set

1. 11 𝑥 + 10 = 132

2. 8 2x + 9 = 56

3. 17 − 5 2x − 9 = − −6x + 10 + 4

4. 5 3x + 4 − 2x = 7x − 3 −2x + 11

5. 3 2𝑥 − 14 + 𝑥 = 15 − (−9𝑥 − 5)

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Unit 3A Lesson 13-17 Classwork - Chenango Forks Middle … 3A Lesson 13-17...

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