## Engage NY Eureka Math 8th Grade Module 4 Lesson 2 Answer Key

### Eureka Math Grade 8 Module 4 Lesson 2 Exercise Answer Key

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.

Exercise 1.
The sum of a number and four times the number
Let x be a number; then, x+4x is a linear expression.

Exercise 2.
The product of five and a number
Let x be a number; then, 5x is a linear expression.

Exercise 3.
Multiply six and the reciprocal of the quotient of a number and seven.
Let x be a number; then, 6∙$$\frac{7}{x}$$ is a nonlinear expression. The expression is nonlinear because the number
$$\frac{7}{x}$$=7∙$$\frac{1}{x}$$=7∙x-1. The exponent of the x is the reason it is not a linear expression.

Exercise 4.
Twice a number subtracted from four times a number, added to 15
Let x be a number; then, 15+(4x-2x) is a linear expression.

Exercise 5.
The square of the sum of six and a number
Let x be a number; then,(x+6)2 is a nonlinear expression. When you multiply(x+6)2, you get x2+12x+36. The x2 is the reason it is not a linear expression.

Exercise 6.
The cube of a positive number divided by the square of the same positive number
Let x be a number; then, $$\frac{x^{3}}{x^{2}}$$ is a nonlinear expression. However, if you simplify the expression to x, then it is linear.

Exercise 7.
The sum of four consecutive numbers
Let x be the first number; then, x+(x+1)+(x+2)+(x+3) is a linear expression.

Exercise 8.
Four subtracted from the reciprocal of a number
Let x be a number; then, $$\frac{1}{x}$$-4 is a nonlinear expression. The term $$\frac{1}{x}$$ is the same as x-1, which is why this expression is not linear. It is possible that a student may let x be the reciprocal of a number, $$\frac{1}{x}$$, which would make the expression linear.

Exercise 9.
Half of the product of a number multiplied by itself three times
Let x be a number; then, $$\frac{1}{2}$$∙x∙x∙x is a nonlinear expression. The term $$\frac{1}{2}$$∙x∙x∙x is the same as$$\frac{1}{2}$$ x3, which is why this expression is not linear.

Exercise 10.
The sum that shows how many pages Maria read if she read 45 pages of a book yesterday and $$\frac{2}{3}$$ of the remaining pages today
Let x be the number of remaining pages of the book; then, 45+$$\frac{2}{3}$$ x is a linear expression.

Exercise 11.
An admission fee of $10 plus an additional$2 per game
Let x be the number of games; then, 10+2x is a linear expression.

Exercise 12.
Five more than four times a number and then twice that sum
Let x be the number; then, 2(4x+5) is a linear expression.

### Eureka Math Grade 8 Module 4 Lesson 2 Problem Set Answer Key

Students practice writing expressions and identifying them as linear or nonlinear.

Write each of the following statements as a mathematical expression. State whether the expression is linear or nonlinear. If it is nonlinear, then explain why.

Question 1.
A number decreased by three squared
Let x be a number; then, x-32 is a linear expression.

Question 2.
The quotient of two and a number, subtracted from seventeen
Let x be a number; then, 17-$$\frac{2}{x}$$ is a nonlinear expression. The term $$\frac{2}{x}$$ is the same as 2∙$$\frac{1}{x}$$ and $$\frac{1}{x}$$=x-1, which is why it is not linear.

Question 3.
The sum of thirteen and twice a number
Let x be a number; then, 13+2x is a linear expression.

Question 4.
5.2 more than the product of seven and a number
Let x be a number; then, 5.2+7x is a linear expression.

Question 5.
The sum that represents the number of tickets sold if 35 tickets were sold Monday, half of the remaining tickets were sold on Tuesday, and 14 tickets were sold on Wednesday
Let x be the remaining number of tickets; then, 35+$$\frac{1}{2}$$ x+14 is a linear expression.

Question 6.
The product of 19 and a number, subtracted from the reciprocal of the number cubed
Let x be a number; then, $$\frac{1}{x^{3}}$$ -19x is a nonlinear expression. The term $$\frac{1}{x^{3}}$$ is the same as x-3, which is why it is not linear.

Question 7.
The product of 15 and a number, and then the product multiplied by itself four times
Let x be a number; then,(15x)4 is a nonlinear expression. The expression can be written as 154∙x4. The exponent of 4 with a base of x is the reason it is not linear.

Question 8.
A number increased by five and then divided by two
Let x be a number; then, $$\frac{x+5}{2}$$ is a linear expression.

Question 9.
Eight times the result of subtracting three from a number
Let x be a number; then, 8(x-3) is a linear expression.

Question 10.
The sum of twice a number and four times a number subtracted from the number squared
Let x be a number; then, x2-(2x+4x) is a nonlinear expression. The term x2 is the reason it is not linear.

Question 11.
One-third of the result of three times a number that is increased by 12
Let x be a number; then, $$\frac{1}{3}$$ (3x+12) is a linear expression.

Question 12.
Five times the sum of one-half and a number
Let x be a number; then, 5($$\frac{1}{2}$$+x) is a linear expression.

Question 13.
Three-fourths of a number multiplied by seven
Let x be a number; then, $$\frac{3}{4}$$x∙7 is a linear expression.

Question 14.
The sum of a number and negative three, multiplied by the number
Let x be a number; then, (x+(-3))x is a nonlinear expression because (x+(-3))x=x2-3x after using the distributive property. It is nonlinear because the power of x in the term x2 is greater than 1.

Question 15.
The square of the difference between a number and 10
Let x be a number; then, (x-10)2 is a nonlinear expression because (x-10)2=x2-20x+100. The term x2 is a positive power of x>1; therefore, this is not a linear expression.

### Eureka Math Grade 8 Module 4 Lesson 2 Exit Ticket Answer Key

Write each of the following statements as a mathematical expression. State whether the expression is a linear or nonlinear expression in x.

Question 1.
Seven subtracted from five times a number, and then the difference added to nine times a number
Let x be a number; then, 15∙$$\frac{1}{x}$$-3x. The expression is a nonlinear expression.
Let x be a number; then, $$\frac{1}{2}$$ (2+x3). The expression is a nonlinear expression.