Examples with solutions for Additional Arithmetic Rules: Using variables

Exercise #1

x(3x+4y)=? x-(3x+4y)=\text{?}

Video Solution

Step-by-Step Solution

First, we open the parentheses and change the sign accordingly.

Since there are only positive numbers within the parentheses, multiplying by a negative will give us negative numbers:

x3x4y= x-3x-4y=

Now we group the X factors:

x3x=2x x-3x=-2x

We obtain:

2x4y -2x-4y

Answer

2x4y -2x-4y

Exercise #2

10y(5y+3z)=? 10y-(5y+3z)=\text{?}

Video Solution

Step-by-Step Solution

First, we open the parentheses and change the sign accordingly.

Since there are only positive numbers inside the parentheses, multiplying by the minus will give us negative numbers:

10y5y3z= 10y-5y-3z=

Now we group the Y factors:

10y5y=5y 10y-5y=5y

Now we obtain:

5y3z 5y-3z

Answer

5y3z 5y-3z

Exercise #3

7x(4b+3x)=? 7x-(4b+3x)=\text{?}

Video Solution

Step-by-Step Solution

First, we open the parentheses and change the sign accordingly.

Since there are only positive numbers inside the parentheses, multiplying by minus will give us negative numbers:

7x4b3x= 7x-4b-3x=

Now we group the X factors:

7x3x=4x 7x-3x=4x

Now we obtain:

4x4b 4x-4b

Answer

4x4b 4x-4b

Exercise #4

x(yx)=? x-(y-x)=\text{?}

Video Solution

Step-by-Step Solution

First, we address the parenthesis:

Remember that:

When we multiply a positive number by a negative number, the result will be negative.

When we multiply a negative number by a negative number, the result will be positive.

We obtain the following:

xy+x= x-y+x=

We join the x coefficients:

x+x=2x x+x=2x

Lastly we obtain:

2xy 2x-y

Answer

2xy 2x-y

Exercise #5

2a+3b(4b3a)=? 2a+3b-(4b-3a)=\text{?}

Video Solution

Step-by-Step Solution

We begin by addressing the parenthesis first:

Remember that:

When we multiply a positive number by a negative number, the result will be negative.

When we multiply a negative number by a negative number, the result will be positive.

Hence we obtain the following calculation:

2a+3b4b+3a= 2a+3b-4b+3a=

We join together the a coefficients:

2a+3a=5a 2a+3a=5a

We then join together the b coefficients:

3b4b=b 3b-4b=-b

We obtain the following:

5ab 5a-b

Answer

5ab 5a-b

Exercise #6

3m14n(7m3n)=? 3m-14n-(7m-3n)=\text{?}

Video Solution

Step-by-Step Solution

We begin by addressing the parenthesis first:

Remember that:

When we multiply a positive number by a negative number the result will be negative.

When we multiply a negative number by a negative number the result will be positive.

Thus we obtain the following equation:

3m14n7m+3n= 3m-14n-7m+3n=

Next we join the m coefficients:

3m7m=4m 3m-7m=-4m

We then join the n coefficients:14n+3n=11n -14n+3n=-11n

Finally we obtain:

4m11n -4m-11n

Answer

4m11n -4m-11n

Exercise #7

12z+3m(mz)=? 12z+3m-(m-z)=\text{?}

Video Solution

Step-by-Step Solution

We begin by addressing the parenthesis:

Remember that:

When we multiply a positive number by a negative number, the result will be negative.

When we multiply a negative number by a negative number, the result will be positive.

Hence we obtain the following calculation:

12z+3mm+z= 12z+3m-m+z=

Next we join together the z coefficients:

12z+z=13z 12z+z=13z

We then join together the m coefficients:

3mm=2m 3m-m=2m

Finally we obtain the following:

13z+2m 13z+2m

Answer

13z+2m 13z+2m

Exercise #8

a+b(ab)=? a+b-(a-b)=\text{?}

Video Solution

Step-by-Step Solution

We begin by addressing the parenthesis first:

Remember that:

When we multiply a positive number by a negative number the result will be negative.

When we multiply a negative number by a negative number the result will be positive.

Hence we obtain the following calculation:

a+ba+b= a+b-a+b=

Next we join together the a coefficients:

aa=0 a-a=0

We then join together the b coefficients:

b+b=2b b+b=2b

We obtain the following:

0+2b=2b 0+2b=2b

Answer

2b 2b

Exercise #9

x:(xy)=? x:(x\cdot y)=\text{?}

Video Solution

Step-by-Step Solution

Let's write the expression as a fraction:

xx×y= \frac{x}{x\times y}=

We'll reduce between the x in the numerator and denominator and get:

1y \frac{1}{y}

Answer

1y \frac{1}{y}

Exercise #10

3m12n/(7m4n)=? 3m\cdot12n/(7m\cdot4n)=\text{?}

Video Solution

Step-by-Step Solution

Let's write the exercise as a fraction:

3m×12n7m×4n= \frac{3m\times12n}{7m\times4n}=

Let's reduce between the m in the numerator and the n in the denominator:

3×127×4= \frac{3\times12}{7\times4}=

Let's write the 12 in the numerator of the fraction as a smaller multiplication exercise:

3×4×37×4= \frac{3\times4\times3}{7\times4}=

Let's reduce between the 4 in the numerator and the denominator:

3×37=97=127 \frac{3\times3}{7}=\frac{9}{7}=1\frac{2}{7}

Answer

127 1\frac{2}{7}

Exercise #11

5x2/(3y20x)=? 5x^2/(3y\cdot20x)=\text{?}

Video Solution

Step-by-Step Solution

Let's write the exercise as a fraction:

5x23y×20x= \frac{5x^2}{3y\times20x}=

We'll factor the numerator of the fraction into a multiplication exercise:

5x×x3y×20x= \frac{5x\times x}{3y\times20x}=

Let's write the 20 in the denominator of the fraction as a smaller multiplication exercise:

5x×x3y×4×5x= \frac{5x\times x}{3y\times4\times5x}=

We'll cancel out the 5x in both the numerator and denominator of the fraction:

x3y×4= \frac{x}{3y\times4}=

Let's multiply the denominator of the fraction:

x12y \frac{x}{12y}

Answer

x12y \frac{x}{12y}

Exercise #12

12a/(7x4b)=? 12a/(7x\cdot4b)=\text{?}

Video Solution

Step-by-Step Solution

Let's begin by writing the exercise as a fraction:

12a7x×4b= \frac{12a}{7x\times4b}=

Next we'll factor the numerator of the fraction into a smaller multiplication exercise:

4×3a7x×4b= \frac{4\times3a}{7x\times4b}=

Let's now reduce the 4 in both the numerator and denominator of the fraction:

3a7xb \frac{3a}{7xb}

Answer

3a7xb \frac{3a}{7xb}

Exercise #13

a:4a=? a:\frac{4}{a}=\text{?}

Video Solution

Step-by-Step Solution

Let's flip the fraction to get a multiplication exercise:

a×a4= a\times\frac{a}{4}=

We'll add the a to the numerator of the fraction:

a×a4=a24 \frac{a\times a}{4}=\frac{a^2}{4}

Answer

a24 \frac{a^2}{4}

Exercise #14

3x(7x+3y)=? 3x-(7x+3y)=\text{?}

Video Solution

Step-by-Step Solution

First, we open the parentheses and change the sign accordingly.

Since there are only positive numbers within the parentheses, multiplying by the minus will give us negative numbers:

3x7x3y= 3x-7x-3y=

Now we group the X factors:

3x7x=4x 3x-7x=-4x

Now we obtain:

4x3y -4x-3y

Answer

4x3y -4x-3y

Exercise #15

8y+4x(13x+8y)=? 8y+4x-(13x+8y)=\text{?}

Video Solution

Step-by-Step Solution

First, we open the parentheses and change the sign accordingly.

Since there are only positive numbers inside the parentheses, multiplying by negative will give us negative numbers:

8y+4x13x8y= 8y+4x-13x-8y=

Now we group the X factors:

4x13x=9x 4x-13x=-9x

Now we group the Y factors:

8y8y=0 8y-8y=0

Now we obtain:

9x -9x

Answer

9x -9x

Exercise #16

a+b(2c+b)=? a+b-(2c+b)=\text{?}

Video Solution

Step-by-Step Solution

First, we open the parentheses and change the sign accordingly.

Since there are only positive numbers inside the parentheses, multiplying by negative will give us negative numbers:

a+b2cb= a+b-2c-b=

Now we group the b factors:

bb=0 b-b=0

Now we obtain:

a2c a-2c

Answer

a2c a-2c

Exercise #17

12x(13x+4y)=? 12x-(13x+4y)=\text{?}

Video Solution

Step-by-Step Solution

First, we open the parentheses and change the sign accordingly.

Since there are only positive numbers inside the parentheses, multiplying by minus will give us negative numbers:

12x13x4y= 12x-13x-4y=

Now we group the x factors:

12x13x=1x 12x-13x=-1x

Now we obtain:

1x4y -1x-4y

Answer

x4y -x-4y

Exercise #18

13x+4y(6x3y)=? 13x+4y-(6x-3y)=\text{?}

Video Solution

Step-by-Step Solution

To begin with we address the parenthesis:

Remember that:

When we multiply a positive number by a negative number, the result will be negative.

When we multiply a negative number by a negative number, the result will be positive.

Thus we obtain the following:

13x+4y6x+3y= 13x+4y-6x+3y=

We then join the x coefficients:

13x6x=7x 13x-6x=7x

We join the y coefficients:

4y+3y=7y 4y+3y=7y

Lastly we obtain:

7x+7y 7x+7y

Answer

7x+7y 7x+7y

Exercise #19

312a+1214b(114a212b)=? 3\frac{1}{2}a+12\frac{1}{4}b-(1\frac{1}{4}a-2\frac{1}{2}b)=\text{?}

Video Solution

Step-by-Step Solution

We begin by addressing the parenthesis:

Remember that:

When we multiply a positive number by a negative number, the result will be negative.

When we multiply a negative number by a negative number, the result will be positive.

We obtain the following equation:

312a+1214b114a+212b= 3\frac{1}{2}a+12\frac{1}{4}b-1\frac{1}{4}a+2\frac{1}{2}b=

Next we join the a coefficients:

312a114a=214a 3\frac{1}{2}a-1\frac{1}{4}a=2\frac{1}{4}a

We then join the b coefficients:

1214b+212b=1434b 12\frac{1}{4}b+2\frac{1}{2}b=14\frac{3}{4}b

Lastly we obtain:

214a+1434b= 2\frac{1}{4}a+14\frac{3}{4}b=

Answer

214a+1434b 2\frac{1}{4}a+14\frac{3}{4}b

Exercise #20

a+b+c(abc)=? a+b+c-(a-b-c)=\text{?}

Video Solution

Step-by-Step Solution

Let's address the expression in parentheses.

Let's remember that:

When we multiply a negative number by a negative number, the result will be positive.

When we multiply a positive number by a negative number, the result will be negative.

Now we get:

a+b+ca+b+c= a+b+c-a+b+c=

Now let's combine the a terms:

aa=0 a-a=0

Now let's combine the b terms:

b+b=2b b+b=2b

Now let's combine the c terms:

c+c=2c c+c=2c

And we get:

0+2b+2c=2b+2c 0+2b+2c=2b+2c

Answer

2b+2c 2b+2c