Solving Equations by Simplifying Like Terms

Simplifying Like Terms in an Equation

When solving equations, simplifying like terms—terms with the same variable and exponent—makes the equation easier to solve by consolidating similar elements. Simplify the like terms in an equation involves combining the elements that belong to the same group. In other words: in all first-degree equations with one unknown, there are elements that belong to the group of unknowns (variables) and elements that belong to the group of numbers. The goal is to unite all the elements of each of the mentioned groups into respective sides to thus arrive at the result of the equation.

In order to so we need to follow these two steps:

Identify Like Terms: Locate terms with identical variable parts on each side of the equation.
Combine Terms: Add or subtract coefficients of like terms to simplify each side.

For example

$X+2X=5+1$

In this equation, we can clearly see that the elements $X$ and $2X$ belong to the group of unknowns, and therefore, we can combine them.

Conversely, the elements $5$ and $1$ belong to the group of numbers and thus can also be combined.

$3X=6$

$X=2$

The result of the equation is $2$ .

Detailed explanation

Start practice

Test yourself on simplifying and combining like terms!

$$ 7x+4x+5x=0 $$

$x=\text{?}$

Practice more now

Below, we provide you with some examples where we apply this method.

Example 1

$6X-1=5X+5$

In this equation, we can clearly see that the elements $6X$ and $5X$ belong to the group of variables, and therefore, we can combine them.

Conversely, the elements $(-1)$ and $5$ belong to the group of numbers, and thus they can also be combined.

$6X-5X=5+1$

$X=6$

The result of the equation is $6$ .

Exercises on Equations by Simplifying Like Terms

Exercise 2

Assignment

$7a+8b+4a+9b=\text{?}$

Solution

We arrange the corresponding elements

$7a+4a+8b+9b=\text{?}$

We add accordingly

$11a+8b+9b=\text{?}$

$11a+17b$

Answer

$11a+17b$

Join Over 30,000 Students Excelling in Math!

Endless Practice, Expert Guidance - Elevate Your Math Skills Today

Test your knowledge

Question 1

$$ 7m+3m-40m=0 $$

$m=\text{?}$

Question 2

$$ 2a+3a+45a=0 $$

$a=\text{?}$

Question 3

$$ x+x=8 $$

Exercise 3

Assignment:

$3x+4x+7+2=\text{?}$

Solution

We add the corresponding elements

$7x+7+2=\text{?}$

$\text{7x+9}$

Answer

$\text{7x+9}$

Exercise 4

Assignment

$18x-7+4x-9-8x=\text{?}$

Solution

We input the corresponding elements

$18x+4x-8x-7-9=\text{?}$

We solve accordingly

$22x-8x-7-9=\text{?}$

$14x-7-9=\text{?}$

Answer

$14x-16$

Do you know what the answer is?

Question 1

$$ -16+a=-17 $$

Question 2

$$ 2+4y-2y=4 $$

Question 3

$$ 8-b=6 $$

Exercise 5

Assignment

$7.3\cdot4a+2.3+8a=\text{?}$

Solution

First, we solve the multiplication exercise

$29.2a+2.3+8a=$

We arrange the terms accordingly

$29.2a+8a+2.3=$

We add the terms accordingly

$37.2a+2.3$

Answer

$37.2a+2.3$

Exercise 6

Assignment

$\frac{3}{8}a+\frac{14}{9}b+1\frac{1}{9}b+\frac{6}{8}a=\text{?}$

Solution

We input the corresponding elements

$\frac{3}{8}a+\frac{6}{8}a+\frac{14}{9}b+1\frac{1}{9}b=$

We add accordingly

$\frac{3+6}{8}a+\frac{14}{9}b+1\frac{1}{9}b=$

We convert the mixed number into an improper fraction

$\frac{3+6}{8}a+\frac{14}{9}b+\frac{10}{9}b=$

We add accordingly

$\frac{9}{8}a+\frac{14+10}{9}b=$

$\frac{9}{8}a+\frac{24}{9}b=$

We convert the improper fractions into mixed numbers

$1\frac{1}{8}a+\frac{24}{9}b=$

$1\frac{1}{8}a+2\frac{6}{9}b$

Answer

$1\frac{1}{8}a+2\frac{6}{9}b$

Check your understanding

Question 1

Find the value of the parameter X

$$ x+3-8x=4+3-x $$

Question 2

Solve for X:

$$ -5x+20-3x=40+2-6x $$

Question 3

Solve for x:

$$ 5+x=3 $$

Examples with solutions for Simplifying and Combining Like Terms

Exercise #1

$7x+4x+5x=0$

$x=\text{?}$

Video Solution

Step-by-Step Solution

Let's combine all the x terms together:

$7x+4x+5x=11x+5x=16x$

The resulting equation is:

$16x=0$

Now let's divide both sides by 16:

$\frac{16x}{16}=\frac{0}{16}$

$x=\frac{0}{16}=0$

Answer

$0$

Exercise #2

$8-b=6$

Video Solution

Step-by-Step Solution

We will move terms so that on the left side of the equation, minus b remains

And to the right side, we will move 8 and make sure to keep the plus and minus signs accordingly:

$-b=6-8$

We will subtract accordingly:

$-b=-2$

We will divide both sides by minus 1 and be careful with the plus and minus signs when dividing by a negative:

$\frac{-b}{-1}=\frac{-2}{-1}$

$b=2$

Answer

$2$

Exercise #3

Solve for x:

$8(-2-x)=16$

Video Solution

Step-by-Step Solution

First, we divide both sections by 8:

$\frac{8(-2-x)}{8}=\frac{16}{8}$

Keep in mind that the 8 in the fraction of the left section is reduced, so the equation we get is:

$-2-x=2$

We move the minus 2 to the right section and maintain the plus and minus signs accordingly:

$-x=2+2$

$-x=4$

We divide both sides by minus 1 and maintain the plus and minus signs accordingly when we divide:

$\frac{-x}{-1}=\frac{4}{-1}$

$x=-4$

Answer

-4

Exercise #4

Solve for x:

$5+x=3$

Video Solution

Step-by-Step Solution

We will rearrange the equation so that x remains on the left side and we will move similar elements to the right side.

Remember that when we move a positive number, it will become a negative number, so we will get:

$x=3-5$

$x=-2$

Answer

-2

Exercise #5

Solve for x:

$-9-x=3+2x$

Video Solution

Step-by-Step Solution

To solve the equation, we will move similar elements to one side.

On the right side, we place the elements with X, while in the left side we place the elements without X.

Remember that when we move sides, the plus and minus signs change accordingly, so we get:

$-9-3=2x+x$

We calculate both sides: $-12=3x$

Finally, divide both sides by 3:

$-\frac{12}{3}=\frac{3x}{3}$

$-4=x$

Answer

-4

Start practice