# Category: Algebra

## Algebra Chapter 16 Lesson 2 Set 1 Problem #3

This is an error in the book by Master Books (ISBN: 978-0890519851) – the book with the blue cover showing the world.

The problem is discussing hens and rabbits with a total of 30 heads and 86 feet.

ERROR: The solution manual is referring to the wrong problem which is related to money.

**Here is how to solve. Simultaneous equations are covered in Chapter 7. So if you forget how to solve these, refer back to chapter 7. **

heads = # hens + # rabbits

feet = 2 x # hens + 4x number of rabbits

So let h be the number of hens and r be the number of rabbits.

We can re-write the above as

heads = 30 = h + r

feet = 86 = 2h + 4r

30=h+r

So r = 30-h

Substitute this equation into the second equation to get

86 = 2h + 4r

86 = 2h + 4(30-h) = 2h + 120 – 4h = -2h +120

Rearrange to get

2h = 120-86 = 34

h = 17 (number of hens)

Now we can find the rabbits are

R = 30-h = 30-17 = 13

**Checking our work**

13 + 17 = 30 (total number of heads)

2 x 17 + 4 x 13 = 86 which are the number of feet.

Correct!

## Algebra Ch13 Test A Problem #13

**Question from Grace:**

We are using this test A as a practice test. We can’t figure out how to solve this.

**Answer from Dr. Callahan:**

Grace,

To solve this one, you take the square root of each side

+/- SQRT ( 16x^2 + 8x +1 ) = +/- SQRT (9)

The left side needs a little work to do this. Best to see if we can factor

16x^2 + 8x +1

As a hint, you know your factors would need to be the same since you are taking a square root.

So start with the first term 16x^2 – square root of it is 4x. So try that

(4x + ___) (4x + ___)

Then the last term is 1. So the obvious thing that can be multiplied by itself to get 1 is 1. So try

(4x + 1) (4x + 1)

Now when I multiply is out I get

16x^2 + 4x + 4x + 1 = 16x^2 + 8x +1

So that works!!

Now back to the problem

+/- SQRT ( 16x^2 + 8x +1 ) = +/- SQRT (9)

+/- SQRT ((4x + 1) (4x + 1)) = +/- SQRT (9)

Using the + we have

(4x + 1) = 3

And 4x = 2 so x = 1/2 is one answer

Using the – we have

(4x + 1) = -3

4x+1 = -3

4x = -4

x = -1

## Algebra Ch10.2 #3

**Question from Makenna**

**Answer from Dr. Callahan**

Makenna,

You are correct. The 678 was from the older version of the book (which used a flag of 104 x 235. Flags have grown since then. 😉)

Just FYI – the older flag was flown in Detroit on the side of the J.L. Hudson Store – which no longer exists.

https://en.wikipedia.org/wiki/J._L._Hudson_Department_Store_and_Addition

## Algebra Ch5.6 #15

Ch5.6 #15

As before – all are similar so I will pick the harder one.

**Givens**

AB = CD

AB = 9(x-2)

CD = x + 3(x+5)

Therefore,

9(x-2) = x + 3(x+5)

**Now multiply out and solve**

9x – 18 = x + 3x + 15

9x – x – 3x = 15 + 18

5x = 33

x = 6.6

**Now go back and find the lengths**

AB = 9(x-2) = 9(6.6 – 2) = 41.4

CD = AB = 41.4

CE = x = 6.6

ED = 3(x+5) = 3(6.6+5) = 34.8

## Algebra Ch5.6 #14

Now let’s look at Ch 5.6 #14.

Parts a-c are similar – so will do part c since the harder.

Area = x + 20 = (x-1)4 > I am just stating all things that are equal based on the given.

x+20 = (x-1)4

x+20 = 4x -4 > multiplied out

20 + 4 = 4x – x > moved similar items to same side

24 = 3x

x = 8

**Now we know x, we have to find the sides (ALWAYS GO BACK AND MAKE SURE YOU KNOW WHAT WAS ASKED)**

x -1 = 8 – 1 = 7

4 = 4

## Algebra Ch5.6 #9

For problem 5.6, parts a-d are all pretty much the same. Here I will do part d) since a hair more complex. The steps the same for the others.

Since AB = CD, we can substitute the following

AB is x + 6(x+2)

CD is 2(x+10)

To get

x + 6(x+2) = 2(x+10)

So to find the length of AB and CD, I need to figure out x. So solve the equation.

x + 6(x+2) = 2(x+10)

x + 6x + 12 = 2x + 20 – I just multiplied everything out

x + 6x – 2x = 20 – 12 > I moved all the x terms to the left and all the constants (numbers) to the right

5x = 8 > did the math

x = 8/5 = 1.6

**Now I have x, but the problem asked for the lengths.**

So now we can find the parts

AB = x + 6(x+2) = 1.6 + 6(1.6 + 2) = 1.6 + 6(3.6) = 23.2

AE = x = 1.6

EB = 6(x+2) = 21.6

CD = AB = 23.2

To do the others – follow the same process

## Algebra Ch5.1 #10x

This is an error in the Algebra Solutions Manual

Mary asked,

**Dr. Callahan Answer:**

## Algebra Textbook Evaluations

Look for the following material or heading in the contents.

- Variables
- Exponents
- Order of Operations
- Equations and inequalities
- Word problems (converting words into symbols)
- Real Numbers
- Adding
- Subtracting
- Multiplication
- Division
- Distributive property
- Linear Equations (might be equations in one variable)
- Graphing
- Slope
- Intercepts
- Point-slope and or slope-intercept formulas
- Systems of linear equations (or simultaneous equations)
- Linear inequalities
- Solving
- Graphing
- Absolute Values
- Exponents
- Products
- Divisions
- Scientific notation
- Polynomials
- Quadratic equations
- Factoring
- Radicals

Notes for use: All the material on the above list needs to be covered in algebra. Instead of moving fast, the students should understand these concepts pretty well.

## Algebra Ch2 Test 2B (additional test)

The test in the Teacher Guide for Test 2B on page 211 has an incorrect answer. The answer given in the solution manual is y=4x+1.

The correct answer is y = 4x-3.

Thanks Marybeth for catching this.