## Engage NY Eureka Math Algebra 1 Module 1 Lesson 10 Answer Key

### Eureka Math Algebra 1 Module 1 Lesson 10 Exercise Answer Key

Exercise 1.

a. Consider the statement: “The president of the United States is a United States citizen.”

Is the statement a grammatically correct sentence?

Answer:

Yes

What is the subject of the sentence? What is the verb in the sentence?

Answer:

President

Is

Is the sentence true?

Answer:

Yes

b. Consider the statement: “The president of France is a United States citizen.”

Is the statement a grammatically correct sentence?

Answer:

Yes

What is the subject of the sentence? What is the verb in the sentence?

Answer:

President

Is

Is the sentence true?

Answer:

No

c. Consider the statement: “2+3=1+4.”

This is a sentence. What is the verb of the sentence? What is the subject of the sentence?

Answer:

Equals

2+3

Is the sentence true?

Answer:

Yes

d. Consider the statement: “2+3=9+4.”

Is this statement a sentence? And if so, is the sentence true or false?

Answer:

Yes

False

Hold a general class discussion about parts (c) and (d) of the exercise. Be sure to raise the following points:

→ One often hears the chime that “mathematics is a language.” And indeed it is. For us reading this text, that language is English. (And if this text were written in French, that language would be French, or if this text were written in Korean, that language would be Korean.)

→ A mathematical statement, such as 2+3=1+4, is a grammatically correct sentence. The subject of the sentence is the numerical expression “2+3”, and its verb is “equals” or “is equal to.” The numerical expression “1+4” renames the subject (2+3). We say that the statement is true because these two numerical expressions evaluate to the same numerical value (namely, five).

→ The mathematical statement 2+3=9+4 is also a grammatically correct sentence, but we say it is false because the numerical expression to the left (the subject of the sentence) and the numerical expression to the right do not evaluate to the same numerical value.

(Perhaps remind students of parts (a) and (b) of the exercise: Grammatically correct sentences can be false.)

Recall the definition:

Exercise 2.

Determine whether the following number sentences are true or false.

a. 4+8=10+5

Answer:

False

b. \(\frac{1}{2}\)+\(\frac{5}{8}\)=1.2-0.075

Answer:

True

c. (71•603)•5876=603•(5876•71)

Answer:

True. The commutative and associative properties of multiplication demand these numerical expressions match.

d. 13×175=13×90+85×13

Answer:

True. Notice the right side equals 13×(90+85).

e. (7+9)^{2}=7^{2}+9^{2}

Answer:

False

f. π=3.141

Answer:

False (The value of π is not exactly 3.141.)

g. \(\sqrt{(4+9)}\)=\(\sqrt{4}\)+\(\sqrt{9}\)

Answer:

False

h. \(\frac{1}{2}\)+\(\frac{1}{3}\)=\(\frac{2}{5}\)

Answer:

False

i. \(\frac{1}{2}\)+\(\frac{1}{3}\)=\(\frac{2}{6}\)

Answer:

False

j. \(\frac{1}{2}\)+\(\frac{1}{3}\)=\(\frac{5}{6}\)

Answer:

True

k. 3^{2}+4^{2}=7^{2}

Answer:

False

l. 3^{2}×4^{2}=12^{2}

Answer:

True

m. 3^{2}×4^{3}=12^{6}

Answer:

False

n. 3^{2}×3^{3}=3^{5}

Answer:

True

Exercise 3.

a. Could a number sentence be both true and false?

b. Could a number sentence be neither true nor false?

Answer:

A number sentence has a left-hand numerical expression that evaluates to a single number and has a right-hand numerical expression that also evaluates to a single numerical value. Either these two single values match or they do not. A numerical sentence is thus either true or false (and not both).

Exercise 4.

Which of the following are algebraic equations?

i. 3.1x-11.2=2.5x+2.3

ii. 10π^{4}+3=99π^{2}

iii. π+π=2π

iv. \(\frac{1}{2}\)+\(\frac{1}{2}\)=\(\frac{2}{4}\)

v. 79π^{3}+70π^{2}-56π+87=\(\frac{60 \pi+29928}{\pi^{2}}\)

Answer:

All of them are all algebraic equations.

b. Which of them are also number sentences?

Answer:

Numbers (ii), (iii), (iv), and (v). (Note that the symbol π has a value that is already stated or known.)

c. For each number sentence, state whether the number sentence is true or false.

Answer:

(ii) False, (iii) True, (iv) False, (v) False. Note that (ii) and (v) are both very close to evaluating to true. Some calculators may not be able to discern the difference. Wolfram Alpha’s web-based application can be used to reveal the differences.

Exercise 5.

When algebraic equations contain a symbol whose value has not yet been determined, we use analysis to determine whether:

a. The equation is true for all the possible values of the variable(s), or

b. The equation is true for a certain set of the possible value(s) of the variable(s), or

c. The equation is never true for any of the possible values of the variable(s).

For each of the three cases, write an algebraic equation that would be correctly described by that case. Use only the variable, x, where x represents a real number.

Answer:

a. 2(x+3)=2x+6; by the distributive property, the two expressions on each side of the equal sign are algebraically equivalent; therefore, the equation is true for all possible real numbers, x.

b. x+5=11; this equation is only a true number sentence if x=6. Any other real number would make the equation a false number sentence.

c. x^{2}=-1; there is no real number x that could make this equation a true number sentence.

Share and discuss some possible answers for each.

Eureka Math Algebra 1 Module 1 Lesson 10 Example Answer Key

Example 1.

Consider the following scenario.

Julie is 300 feet away from her friend’s front porch and observes, “Someone is sitting on the porch.”

Given that she did not specify otherwise, we would assume that the “someone” Julie thinks she sees is a human. We cannot guarantee that Julie’s observational statement is true. It could be that Julie’s friend has something on the porch that merely looks like a human from far away. Julie assumes she is correct and moves closer to see if she can figure out who it is. As she nears the porch, she declares, “Ah, it is our friend, John Berry.”

→ Often in mathematics, we observe a situation and make a statement we believe to be true. Just as Julie used the word “someone”, in mathematics we use variables in our statements to represent quantities not yet known. Then, just as Julie did, we “get closer” to study the situation more carefully and find out if our “someone” exists and, if so, “who” it is.

→ Notice that we are comfortable assuming that the “someone” Julie referred to is a human, even though she did not say so. In mathematics we have a similar assumption. If it is not stated otherwise, we assume that variable symbols represent a real number. But in some cases, we might say the variable represents an integer or an even integer or a positive integer, for example.

→ Stating what type of number the variable symbol represents is called stating its domain.

Exercise 6.

Name a value of the variable that would make each equation a true number sentence.

Here are several examples of how we can name the value of the variable.

__Let w=-2__. Then w^{2}=4 is true.

w^{2}=4 is true __when w=-2__

w^{2}=4 is true __if w=-2__

w^{2}=4 is true __for w=-2 and w=2__.

There might be more than one option for what numerical values to write. (And feel free to write more than one possibility.)

Warning: Some of these are tricky. Keep your wits about you!

a. Let ____. Then, 7+x=12 is true.

Answer:

x=5

b. Let _____ . Then, 3r+0.5=\(\frac{37}{2}\) is true.

Answer:

r=6

c. m^{3}=-125 is true for ____.

Answer:

m=-5

d. A number x and its square, x^{2}, have the same value when .

Answer:

x=1 or when x=0

e. The average of 7 and n is -8 if .

Answer:

n=-23

f. Let ___. Then, 2a=a+a is true.

Answer:

a= any real number

g. q+67=q+68 is true for ___.

Answer:

There is no value one can assign to q to turn this equation into a true statement.

Eureka Math Algebra 1 Module 1 Lesson 10 Exit Ticket Answer Key

Question 1.

Consider the following equation, where a represents a real number: \(\sqrt{a+1}\)=\(\sqrt{a}\)+1. Is this statement a number sentence? If so, is the sentence true or false?

Answer:

No, it is not a number sentence because no value has been assigned to a. Thus, it is neither true nor false.

Question 2.

Suppose we are told that b has the value 4. Can we determine whether the equation below is true or false? If so, say which it is; if not, state that it cannot be determined. Justify your answer.

\(\sqrt{b+1}\)=\(\sqrt{\boldsymbol{b}}\) + 1

Answer:

False. The left-hand expression has value \(\sqrt{4+1}\)=\(\sqrt{5}\) and the right-hand expression has value 2+1=3. These are not the same value.

Question 3.

For what value of c is the following equation true?

\(\sqrt{c+1}\)=\(\sqrt{c}\) + 1

\(\sqrt{c+1}\)=\(\sqrt{c}\) +1, if we let c = 0.

Eureka Math Algebra 1 Module 1 Lesson 10 Problem Set Answer Key

Determine whether the following number sentences are true or false.

Question 1.

18+7=\(\frac{50}{2}\)

Answer:

true

Question 2.

3.123=9.369 \(\frac{1}{3}\)

Answer:

true

Question 3.

(123 + 54)∙4=123+(54∙4)

Answer:

false

Question 4.

5^{2}+12^{2}=13^{2}

Answer:

true

Question 5.

(2×2)^{2}=\(\sqrt{256}\)

Answer:

true

Question 6.

\(\frac{4}{3}\)=1.333

Answer:

false

In the following equations, let x=-3 and y=\(\frac{2}{3}\). Determine whether the following equations are true, false, or neither true nor false.

Question 7.

xy=-2

Answer:

true

Question 8.

x+3y=-1

Answer:

true

Question 9.

x+z=4

Answer:

Neither true nor false

Question 10.

9y=-2x

Answer:

true

Question 11.

\(\frac{y}{x}\)=-2

Answer:

false

Question 12.

\(\frac{-\frac{2}{x}}{y}\)=-1

Answer:

false

For each of the following, assign a value to the variable, x, to make the equation a true statement.

Question 13.

(x^{2}+5)(3+x^{4} )(100x^{2}-10)(100x^{2}+10)=0 for .

Answer:

x=\(\frac{1}{\sqrt{10}}\) or x=-\(\frac{1}{\sqrt{10}}\)

Question 14.

\(\sqrt{(x+1)(x+2)}\)=\(\sqrt{20}\) for ___.

Answer:

x=3 or x=-6.

Question 15.

(d+5)^{2}=36 for ___.

Answer:

d=1 or d=-11

Question 16.

(2z+2)(z^{5}-3)+6=0 for ___ .

Answer:

z=0 seems the easiest answer.

Question 17.

\(\frac{1+x}{1+x^{2}}\) = \(\frac{3}{5}\) for ____.

Answer:

x=2 works.

Question 18.

\(\frac{1+x}{1+x^{2}}\)=\(\frac{2}{5}\) for ___ .

Answer:

x=3 works, and so does x=-\(\frac{1}{2}\).

Question 19.

The diagonal of a square of side length L is 2 inches long when ___.

Answer:

L=\(\sqrt{2}\) inches

Question 20.

(T-\(\sqrt{3}\))^{2}=T^{2}+3 for _____.

Answer:

T=0

Question 21.

\(\frac{1}{x}\)=\(\frac{x}{1}\) if ___.

Answer:

x=1 and also if x=-1

Question 22.

(2+(2-(2+(2-(2+r)))))=1 for _____.

Answer:

r= -1

Question 23.

x+2=9

Answer:

for x=7

Question 24.

x+2^{2}=-9

Answer:

for x=-13

Question 25.

-12t=12

Answer:

for t=-1

Question 26.

12t=24

Answer:

for t=2

Question 27.

\(\frac{1}{b-2}\)=\(\frac{1}{4}\)

Answer:

for b=6

Question 28.

\(\frac{1}{2 b-2}\)=\(\frac{1}{4}\)

Answer:

for b= -1

Question 29.

\(\sqrt{x}\)+\(\sqrt{5}\)=\(\sqrt{x+5}\)

Answer:

for x=0

Question 30.

(x-3)^{2}=x^{2}+(-3)^{2}

Answer:

for x=0

Question 31.

x^{2}=-49

Answer:

No real number will make the equation true.

Question 32.

\(\frac{2}{3}\)+\(\frac{1}{5}\)=\(\frac{3}{x}\)

Answer:

for x=\(\frac{45}{13}\)

Fill in the blank with a variable term so that the given value of the variable will make the equation true.

Question 33.

___ + 4=12; x=8

Answer:

__x__ + 4=12; x=8

Question 34.

___ +4=12; x=4

Answer:

__2x__ +4=12; x=4

Fill in the blank with a constant term so that the given value of the variable will make the equation true.

Question 35.

___ – 0 = 100; y=25

Answer:

4y- __0__ =100; y=25

Question 36.

4y- ___ =0; y=6

Answer:

4y- __24__ =0; y=6

Question 37.

r + __ =r; r is any real number.

Answer:

r + __0__ = r; r is any real number.

Question 38.

r× ___ = r; r is any real number.

Answer:

r× __1__ = r; r is any real number.

Generate the following:

Answer:

Answers will vary. Sample responses are provided below.

Question 39.

An equation that is always true

Answer:

2x+4=2(x+2)

Question 40.

An equation that is true when x=0

Answer:

x+2=2

Question 41.

An equation that is never true

Answer:

x+3=x+2

Question 42.

An equation that is true when t=1 or t=-1

Answer:

t^{2}=1

Question 43.

An equation that is true when y=-0.5

Answer:

2y+1=0

Question 44.

An equation that is true when z=π

Answer:

\(\frac{z}{\pi}\)=1