Additional Arithmetic Rules: Using variables

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

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:

$12z+3m-m+z=$

Next we join together the z coefficients:

$12z+z=13z$

We then join together the m coefficients:

$3m-m=2m$

Finally we obtain the following:

$13z+2m$

$13z+2m$

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

Let's write the exercise as a fraction:

$\frac{3m\times12n}{7m\times4n}=$

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

$\frac{3\times12}{7\times4}=$

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

$\frac{3\times4\times3}{7\times4}=$

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

$\frac{3\times3}{7}=\frac{9}{7}=1\frac{2}{7}$

$1\frac{2}{7}$

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

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:

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

We join together the a coefficients:

$2a+3a=5a$

We then join together the b coefficients:

$3b-4b=-b$

We obtain the following:

$5a-b$

$5a-b$

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

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:

$a+b-a+b=$

Next we join together the a coefficients:

$a-a=0$

We then join together the b coefficients:

$b+b=2b$

We obtain the following:

$0+2b=2b$

$2b$

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

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:

$10y-5y-3z=$

Now we group the Y factors:

$10y-5y=5y$

Now we obtain:

$5y-3z$

$5y-3z$

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

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:

$x-3x-4y=$

Now we group the X factors:

$x-3x=-2x$

We obtain:

$-2x-4y$

$-2x-4y$

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

Let's write the exercise as a fraction:

$\frac{5x^2}{3y\times20x}=$

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

$\frac{5x\times x}{3y\times20x}=$

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

$\frac{5x\times x}{3y\times4\times5x}=$

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

$\frac{x}{3y\times4}=$

Let's multiply the denominator of the fraction:

$\frac{x}{12y}$

$\frac{x}{12y}$

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

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:

$7x-4b-3x=$

Now we group the X factors:

$7x-3x=4x$

Now we obtain:

$4x-4b$

$4x-4b$

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

Let's write the exercise as a fraction:

$\frac{12a}{7x\times4b}=$

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

$\frac{4\times3a}{7x\times4b}=$

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

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

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

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

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

$a\times\frac{a}{4}=$

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

$\frac{a\times a}{4}=\frac{a^2}{4}$

$\frac{a^2}{4}$

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

Let's write the expression as a fraction:

$\frac{x}{x\times y}=$

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

$\frac{1}{y}$

$\frac{1}{y}$

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

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:

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

Next we join the m coefficients:

$3m-7m=-4m$

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

Finally we obtain:

$-4m-11n$

$$$-4m-11n$

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

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:

$x-y+x=$

We join the x coefficients:

$x+x=2x$

Lastly we obtain:

$2x-y$

$2x-y$

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

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:

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

Now we group the X factors:

$4x-13x=-9x$

Now we group the Y factors:

$8y-8y=0$

Now we obtain:

$-9x$

$-9x$

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

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:

$12x-13x-4y=$

Now we group the x factors:

$12x-13x=-1x$

Now we obtain:

$-1x-4y$

$-x-4y$

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

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:

$a+b+c-a+b+c=$

Now let's combine the a terms:

$a-a=0$

Now let's combine the b terms:

$b+b=2b$

Now let's combine the c terms:

$c+c=2c$

And we get:

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

$2b+2c$

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

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:

$3x-7x-3y=$

Now we group the X factors:

$3x-7x=-4x$

Now we obtain:

$-4x-3y$

$-4x-3y$

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

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:

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

We then join the x coefficients:

$13x-6x=7x$

We join the y coefficients:

$4y+3y=7y$

Lastly we obtain:

$7x+7y$

$7x+7y$

$3\frac{1}{2}a+12\frac{1}{4}b-(1\frac{1}{4}a-2\frac{1}{2}b)=\text{?}$

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:

$3\frac{1}{2}a+12\frac{1}{4}b-1\frac{1}{4}a+2\frac{1}{2}b=$

Next we join the a coefficients:

$3\frac{1}{2}a-1\frac{1}{4}a=2\frac{1}{4}a$

We then join the b coefficients:

$12\frac{1}{4}b+2\frac{1}{2}b=14\frac{3}{4}b$

Lastly we obtain:

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

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

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

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:

$a+b-2c-b=$

Now we group the b factors:

$b-b=0$

Now we obtain:

$a-2c$

$a-2c$