Solving Equations by Simplifying Like Terms

🏆Practice simplifying and combining like terms

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.

For example

A1 - Solving Equations by Simplifying Like Terms

X+2X=5+1 X+2X=5+1

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

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

3X=6 3X=6

X=2 X=2

The result of the equation is 2 2 .


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\( x+x=8 \)

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

Example 1

6X1=5X+5 6X-1=5X+5

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

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

6X5X=5+1 6X-5X=5+1

X=6 X=6

The result of the equation is 6 6 .


Exercises on Equations by Simplifying Like Terms

Exercise 2

Assignment

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

Solution

We arrange the corresponding elements

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

We add accordingly

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

11a+17b 11a+17b

Answer

11a+17b 11a+17b


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Exercise 3

Assignment:

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

Solution

We add the corresponding elements

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

7x+9 \text{7x+9}

Answer

7x+9 \text{7x+9}


Exercise 4

Assignment

18x7+4x98x=? 18x-7+4x-9-8x=\text{?}

Solution

We input the corresponding elements

18x+4x8x79=? 18x+4x-8x-7-9=\text{?}

We solve accordingly

22x8x79=? 22x-8x-7-9=\text{?}

14x79=? 14x-7-9=\text{?}

Answer

14x16 14x-16


Do you know what the answer is?

Exercise 5

Assignment

7.34a+2.3+8a=? 7.3\cdot4a+2.3+8a=\text{?}

Solution

First, we solve the multiplication exercise

29.2a+2.3+8a= 29.2a+2.3+8a=

We arrange the terms accordingly

29.2a+8a+2.3= 29.2a+8a+2.3=

We add the terms accordingly

37.2a+2.3 37.2a+2.3

Answer

37.2a+2.3 37.2a+2.3


Exercise 6

Assignment

38a+149b+119b+68a=?\frac{3}{8}a+\frac{14}{9}b+1\frac{1}{9}b+\frac{6}{8}a=\text{?}

Solution

We input the corresponding elements

38a+68a+149b+119b=\frac{3}{8}a+\frac{6}{8}a+\frac{14}{9}b+1\frac{1}{9}b=

We add accordingly

3+68a+149b+119b= \frac{3+6}{8}a+\frac{14}{9}b+1\frac{1}{9}b=

We convert the mixed number into an improper fraction

3+68a+149b+109b= \frac{3+6}{8}a+\frac{14}{9}b+\frac{10}{9}b=

We add accordingly

98a+14+109b= \frac{9}{8}a+\frac{14+10}{9}b=

98a+249b= \frac{9}{8}a+\frac{24}{9}b=

We convert the improper fractions into mixed numbers

118a+249b= 1\frac{1}{8}a+\frac{24}{9}b=

118a+269b 1\frac{1}{8}a+2\frac{6}{9}b

Answer

118a+269b 1\frac{1}{8}a+2\frac{6}{9}b


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Examples with solutions for Simplifying and Combining Like Terms

Exercise #1

8b=6 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=68 -b=6-8

We will subtract accordingly:

b=2 -b=-2

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

b1=21 \frac{-b}{-1}=\frac{-2}{-1}

b=2 b=2

Answer

2 2

Exercise #2

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

x=? x=\text{?}

Video Solution

Step-by-Step Solution

Let's combine all the x terms together:

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

The resulting equation is:

16x=0 16x=0

Now let's divide both sides by 16:

16x16=016 \frac{16x}{16}=\frac{0}{16}

x=016=0 x=\frac{0}{16}=0

Answer

0 0

Exercise #3

Solve for x:

8(2x)=16 8(-2-x)=16

Video Solution

Step-by-Step Solution

First, we divide both sections by 8:

8(2x)8=168 \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:

2x=2 -2-x=2

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

x=2+2 -x=2+2

x=4 -x=4

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

x1=41 \frac{-x}{-1}=\frac{4}{-1}

x=4 x=-4

Answer

-4

Exercise #4

Solve for x:

5+x=3 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=35 x=3-5

x=2 x=-2

Answer

-2

Exercise #5

Solve for x:

9x=3+2x -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:

93=2x+x -9-3=2x+x

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

Finally, divide both sides by 3:

123=3x3 -\frac{12}{3}=\frac{3x}{3}

4=x -4=x

Answer

-4

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