# Linear and Non-Linear Expressions

Video solutions to help Grade 8 students learn the properties of linear and non-linear expressions in x.

## New York State Common Core Math Module 4, Grade 8, Lesson 2

### Lesson 2 Student Outcomes

• Students know the properties of linear and non-linear expressions in x.
• Students transcribe and identify expressions as linear or non-linear.

Lesson 2 Summary
Linear expressions are sums of constants and products of constants and raised to a power of 0 or 1.
Non-linear expressions are also sums of constants and products of constants and a raised to a power that is not 0 or 1.

Classwork
Discussion:
• Terms are any product of an integer power of x and a constant or just a constant.
• Constants are fixed numbers.
• When a term is the product of a constant(s)and a power of x, the constant is called a coefficient.
• A symbolic statement in x with an equal sign is called an equation in x. The equal sign divides the equation into two parts, the left side and the right side. The two sides are called expressions.
• Linear expressions in x are a special type of expression. Linear expressions are expressions that are sums of constants and products of a constant and x.
• Non-linear expressions are also sums of constants and products of a constant and a power of x. However, non-linear expressions will have a power of that is not equal to 1 or 0.

Example 1:
A linear expression in is an expression where each term is either a constant, an x or a product of a constant and x.
Why is 2x2 + 9x + 5 not a linear expression in x?

Example 2:
Is 4 + 3x5 a linear or non-linear expression in Why or why not?

Example 3:
How many terms does the expression 7x + 9 + 6 + 3x have? What are they?
How many terms with coefficients does the expression 10x + 15 have? What are they?
Is 10x + 15 a linear or non-linear expression in ? Why or why not?

Example 4:
How many terms does the expression 5 + 9x • 7 + 2x9 have? What are they?
How many terms with coefficients does the expression 5 + 9x • 7 + 2x9 have? What are they?
Is 5 + 9x • 7 + 2x9 a linear or non-linear expression in ? Why or why not?

Example 5:
Is 94 + x + 4x-6 - 2 a linear or non-linear expression in ? Why or why not?

Example 6:
Is the expression x1 + 9x - 4 a linear expression in x?

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

1. The sum of a number and four times the number.
2. The product of five and a number.
3. Multiply six and the reciprocal of the quotient of a number and seven.
4. Twice a number subtracted from four times a number, added to 15.
5. The square of the sum of six and a number.
6. The cube of a positive number divided by the square of the same positive number.
7. The sum of four consecutive numbers.
8. Four subtracted from the reciprocal of a number.
9. Half of the product of a number multiplied by itself, three times.
10. The sum that shows how many pages Maria read if she read 45 pages of a book yesterday and 2/3 of the remaining pages today.
11. An admission fee of \$10 plus an additional \$2 per game.
12. Five more than four times a number, then twice that sum.