Associative Property: Using variables

Let's look at the exercise, and we'll see that we have two "regular" numbers and one number with a variable.
Since this is a multiplication exercise, there's no problem multiplying a number with a variable by a number without a variable.

In fact, it's important to remember that a variable attached to a number represents multiplication itself, for example in this case: $11\times x$
Therefore, we can use the distributive property to separate the variable, and come back to it later.
We'll solve the exercise from right to left since it's simpler this way.

$5\times6=30$

Now we'll get the exercise:

$11x\times30=$

We'll put aside the x and add it at the end of the exercise.

Solve the exercise in an organized way to make the solving process easier for ourselves.

It's important to maintain the correct order of solving, meaning first multiply the ones of the first number by the ones of the second number,
then the tens of the first number by the ones of the second number, and so on.

$30\\\times11\\=330$

Don't forget to add the variable at the end, and thus the answer will be:

$330x$

Let's solve the exercise from left to right.

We will combine the left expression in the following way:

$\frac{6+8}{7}x=\frac{14}{7}x=2x$

Now we get:

$2x+3\frac{2}{3}x=5\frac{2}{3}x$

Let's look at the exercise, and we'll see that we have two "regular" numbers and one number with a variable.
Since this is a multiplication exercise, there's no problem multiplying a number with a variable by a number without a variable.

In fact, it's important to remember that a variable attached to a number represents multiplication by itself, for example in this case: $2\times x$
Therefore, we can use the distributive property to separate the variable, and come back to it later.
We'll solve the exercise from left to right.

We'll solve the left exercise by breaking down the decimal number into an addition problem of a whole number and a decimal number as follows:

$2\times(4+0.65)=$

We'll multiply 2 by each term in parentheses:

$(2\times4)+(2\times0.65)=$

We'll solve each of the expressions in parentheses and get:

$8+1.3=9.3$

Now we'll get the exercise:

$9.3\times6.3=$

We'll solve the exercise vertically to make the process easier for ourselves.

It's important to be careful with the proper placement of the exercise, using the decimal point as an anchor.
Then we can multiply in order, first the ones digit of the first number by the ones digit of the second number,
then the tens digit of the first number by the ones digit of the second number, and so on.

$9.3\\\times6.3\\=58.59$

Don't forget to add the variable at the end, and the answer will be:

$58.59x$

According to the order of operations rules, we'll solve the exercise from left to right:

$0.2x+8.6x=8.8x$

We'll break down 8.8 into a smaller addition exercise that will be easier for us to calculate:

$8x+0.8x+0.65x=$

Now we'll use the commutative property since the exercise only involves addition.

Let's focus on the leftmost addition exercise, remembering that:

$0.8=0.80$$$

We'll calculate the following exercise:

$0.80x+0.65x=1.45x$

And finally, we'll get the exercise:

$8x+1.45x=9.45x$

The first step is factorising each of the terms in the exercise into a whole number and its remainder:

$10x+0.1x+5x+0.2x+2x+0.4x=$

Now we'll combine only the whole numbers:

$10x+5x+2x=15x+2x=17x$

Next, we will calculate the remainder:

$0.1x+0.2x+0.4x=0.3x+0.4x=0.7x$

Finally, we are left with the following:

$17x+0.7x=17.7x$

Let's begin by combining the simple fractions into a single multiplication exercise:

$\frac{3\times2}{4\times3}\times2\frac{1}{4}x=$

Let's now proceed to solve the exercise in the numerator and denominator:

$\frac{6}{12}\times2\frac{1}{4}x=$

Finally we'll simplify the simple fraction in order to obtain the following:

$\frac{1}{2}\times2\frac{1}{4}x=1\frac{1}{8}x$

Let's look at the exercise, and we'll see that we have two "regular" numbers and one number with a variable.
Since this is a multiplication exercise, there's no problem multiplying a number with a variable by a number without a variable.

In fact, it's important to remember that a variable attached to a number represents multiplication by itself, for example in this case: $5.2\times x$
Therefore, we can use the distributive property to separate the variable, and come back to it later.
We'll solve the exercise from left to right.

We'll solve the left exercise vertically to avoid confusion and get:

$~~~~~15.6 \\\times~~~~5.2 \\=~81.12$

It's important to be careful with the correct placement of the exercise, where the decimal point serves as an anchor.
Then we can multiply in order, first the ones digit of the first number by the ones digit of the second number,
then the tens digit of the first number by the ones digit of the second number, and so on.

Now we'll get the exercise:

$81.21\times0.3=$

Let's remember that:

$0.3=\text{0}.30$

And we'll get:

$24.336$

Let's not forget to add the variable at the end, and thus the answer will be:

$24.336 x$

This exercise can be solved in order, but to make it easier, the associative property can be used

$-5+(3+4)=$

$-5+7=$

$7-5=2$

We move the fraction to the beginning of the exercise and will place the rest of the exercise in parentheses to make solving the equation easier:

$\frac{a}{4}+(2-2)=$

$\frac{a}{4}+0=\frac{a}{4}$

First, let's find a common denominator for 4, 8, and 6: it's 24.

Now, we'll multiply each fraction by the appropriate number to get:

$\frac{5\times4}{6\times4}x+\frac{7\times3}{8\times3}x+\frac{2\times6}{4\times6}x=$

Let's solve the multiplication exercises in the numerator and denominator:

$\frac{20}{24}x+\frac{21}{24}x+\frac{12}{24}x=$

We'll connect all the numerators:

$\frac{20+21+12}{24}x=\frac{41+12}{24}x=\frac{53}{24}x$

Let's break down the numerator into a smaller addition exercise:

$\frac{48+5}{24}=\frac{48}{24}+\frac{5}{24}=2+\frac{5}{24}=2\frac{5}{24}x$

First, let's convert all mixed fractions to simple fractions:

$\frac{3\times6+5}{6}\times\frac{5\times6+5}{6}\times\frac{1}{3}x=$

Let's solve the exercises with the eight fractions:

$\frac{18+5}{6}\times\frac{30+5}{6}\times\frac{1}{3}x=$

$\frac{23}{6}\times\frac{35}{6}\times\frac{1}{3}x=$

Since the exercise only involves multiplication, we'll combine all the numerators and denominators:

$\frac{23\times35}{6\times6\times3}x=\frac{805}{108}x$

According to the rules of the order of operations, we first eliminate the parentheses.

Remember that a positive times a positive will give a positive result, and a positive times a negative will give a negative result.

Therefore, we obtain:

$4a+5a-2-4a+10=$

Now, we arrange the exercise in a more comfortable way using the substitution property:

$4a+5a-4a+10-2=$

We solve the exercise from left to right, starting by adding the coefficients a:

$4a+5a-4a=9a-4a=5a$

Now we obtain:

$5a+10-2=5a+8$