Solution of an equation by adding/subtracting two sides - Examples, Exercises and Solutions

This method allows us to add or subtract the same element from both sides of the equation without changing the final result, that is, the outcome of the equation will not be affected by the fact that we have added or subtracted the same element from both sides.

Solving Equations by Adding or Subtracting the Same Number from Both Sides

Let's see what the logic of this method is:

José and Isabel, for example, are twin siblings who receive their weekly allowance for the first time.

José and Isabel receive 10 10 euros each, so at this moment they have exactly 10 10 euros per person.

After a month, each has received another 2 2 euros, so now each has 12 12 euros.

We see that adding 2 2 euros to the amount each of them had has not affected the equivalence between them: both still have the same amount of money.

Practice Solution of an equation by adding/subtracting two sides

Exercise #1

Solve the equation and find Y:

20×y+8×27=14 20\times y+8\times2-7=14

Video Solution

Step-by-Step Solution

First, we will put parentheses around the two multiplication exercises:

(20×y)+(8×2)7=14 (20\times y)+(8\times2)-7=14

We solve the exercises within the parentheses:

20y+167=14 20y+16-7=14

We simplify:

20y+9=14 20y+9=14

We move the sections:

20y=149 20y=14-9

20y=5 20y=5

We divide by 20:

y=520 y=\frac{5}{20}

y=55×4 y=\frac{5}{5\times4}

We simplify:

y=14 y=\frac{1}{4}

Answer

14 \frac{1}{4}

Exercise #2

92x×224 ⁣:4=64 92-x\times2-24\colon4=64 Calculate X.

Video Solution

Step-by-Step Solution

First, we solve the multiplication and division exercises, we will put them in parentheses to avoid confusion:

92(x×2)(24 ⁣:4)=64 92-(x\times2)-(24\colon4)=64

922x6=64 92-2x-6=64

Reduce:

862x=64 86-2x=64

Move the sides:

2x=6486 -2x=64-86

2x=22 -2x=-22

Divide by negative 2:

x=222 x=\frac{-22}{-2}

x=11 x=11

Answer

11

Exercise #3

Solve for X:

5x8=10x+22 5x-8=10x+22

Video Solution

Step-by-Step Solution

First, we arrange the two sections so that the right side contains the values with the coefficient x and the left side the numbers without the x

Let's remember to maintain the plus and minus signs accordingly when we move terms between the sections.

First, we move a5x 5x to the right section and then the 22 to the left side. We obtain the following equation:

822=10x5x -8-22=10x-5x

We subtract both sides accordingly and obtain the following equation:

30=5x -30=5x

We divide both sections by 5 and obtain:

6=x -6=x

Answer

6 -6

Exercise #4

What is the missing number?

2312×(6)+? ⁣:7=102 23-12\times(-6)+?\colon7=102

Video Solution

Step-by-Step Solution

First, we solve the multiplication exercise:

12×(6)=72 12\times(-6)=-72

Now we get:

23(72)+x ⁣:7=102 23-(-72)+x\colon7=102

Let's pay attention to the minus signs, remember that a negative times a negative equals a positive.

We multiply them one by one to be able to open the parentheses:

23+72+x ⁣:7=102 23+72+x\colon7=102

We reduce:

95+x:7=102 95+x:7=102

We move the sections:

x:7=10295 x:7=102-95

x:7=7 x:7=7

x7=7 \frac{x}{7}=7

Multiply by 7:

x=7×7=49 x=7\times7=49

Answer

49

Exercise #5

Fill in the missing number:

? ⁣:(6)+[2(9473)]=153 ?\colon(-6)+\lbrack-2(94-73)\rbrack=-153

Video Solution

Step-by-Step Solution

We solve the innermost parentheses:

9473=21 94-73=21

We obtain:

x ⁣:(6)+[2(21)]=153 x\colon(-6)+\lbrack-2(21)\rbrack=-153

We will focus on the parentheses and multiply:

2×21=42 -2\times21=-42

We obtain:

x ⁣:(6)+(42)=153 x\colon(-6)+(-42)=-153

Let's consider the multiplication between minus and plus:

x ⁣:(6)42=153 x\colon(-6)-42=-153

We swap sections:

x ⁣:(6)=153+42 x\colon(-6)=-153+42

x ⁣:(6)=111 x\colon(-6)=-111

x6=111 \frac{x}{-6}=-111

We multiply by minus 6:

x=111×6=666 x=-111\times-6=666

Answer

x=666 x=666

Exercise #1

Solve for X:

18x23x+15x=1 \frac{1}{8}x-\frac{2}{3}x+\frac{1}{5}x=1

Video Solution

Step-by-Step Solution

The common denominator of 8, 3, and 5 is 120.

Now we multiply each numerator by the corresponding number to reach 120 and thus cancel the fractions and obtain the following equation:

(1×x×15)(2×x×40)+(1×x×24)=1×120 (1\times x\times15)-(2\times x\times40)+(1\times x\times24)=1\times120

We multiply the exercises in parentheses accordingly:

15x80x+24x=120 15x-80x+24x=120

We will solve the left side (from left to right) and will obtain:

(15x80x)+24x=120 (15x-80x)+24x=120

65x+24x=120 -65x+24x=120

41x=120 -41x=120

We reduce both sides by 41 -41

41x41=12041 \frac{-41x}{-41}=\frac{120}{-41}

We find that x is equalx=12041 x=-\frac{120}{41}

Answer

12041 -\frac{120}{41}

Exercise #2

Fill in the missing number:

? ⁣:4+8×37=26 ?\colon4+8\times3-7=26

Video Solution

Step-by-Step Solution

First, we place the multiplication exercise in parentheses:

x ⁣:4+(8×3)7=26 x\colon4+(8\times3)-7=26

We solve the multiplication exercise:

x ⁣:4+247=26 x\colon4+24-7=26

We place the subtraction exercise in parentheses:

x ⁣:4+(247)=26 x\colon4+(24-7)=26

We solve the subtraction exercise:

x ⁣:4+17=26 x\colon4+17=26

We swap sections:

x ⁣:4=2617 x\colon4=26-17

x4=9 \frac{x}{4}=9

Multiply by 4

x=4×9=36 x=4\times9=36

Answer

?=36 ?=36

Exercise #3

How much is xequal to?

2542 ⁣:x+18×2 ⁣:4=23 -25-42\colon x+18\times2\colon4=-23

Video Solution

Step-by-Step Solution

First, we will put the multiplication exercise in parentheses:

2542 ⁣:x+(18×2) ⁣:4=23 -25-42\colon x+(18\times2)\colon4=-23

2542 ⁣:x+36 ⁣:4=23 -25-42\colon x+36\colon4=-23

We will put the division exercise in parentheses:

2542 ⁣:x+(36 ⁣:4)=23 -25-42\colon x+(36\colon4)=-23

2542 ⁣:x+9=23 -25-42\colon x+9=-23

We arrange the exercise so that we can simplify:

25+942 ⁣:x=23 -25+9-42\colon x=-23

(25+9)42 ⁣:x=23 (-25+9)-42\colon x=-23

We solve the exercise in parentheses and obtain:

1642 ⁣:x=23 -16-42\colon x=-23

We move the fractions and obtain:

42 ⁣:x=23+16 -42\colon x=-23+16

42 ⁣:x=7 -42\colon x=-7

We multiply by x and obtain:

42=7x -42=-7x

We divide by negative 7:

x=427=6 x=\frac{-42}{-7}=6

Answer

6

Exercise #4

What is the missing number in the equation?

5×17112?14=27.5 5\times17-112-\frac{?}{14}=-27.5

Video Solution

Step-by-Step Solution

We place the multiplication exercise in the parenthesis:

(5×17)112x14=27.5 (5\times17)-112-\frac{x}{14}=-27.5

We solve the multiplication exercise and then subtract:

85112x14=27.5 85-112-\frac{x}{14}=-27.5

27x14=27.5 -27-\frac{x}{14}=-27.5

We swap sections:

x14=27.5+27 -\frac{x}{14}=-27.5+27

x14=12 -\frac{x}{14}=-\frac{1}{2}

We multiply by 14:

x=12×14 -x=-\frac{1}{2}\times14

x=7 -x=-7

We multiply by -1

x=7 x=7

Answer

7 7

Exercise #5

Calculate the missing number in the equation:

(4522)×19+25 ⁣:5+?×3=82 (45-22)\times19+25\colon5+?\times3=82

Video Solution

Step-by-Step Solution

We solve the exercise in parentheses:

23×19+25 ⁣:5+?×3=82 23\times19+25\colon5+?\times3=82

We place in parentheses the multiplication and division exercises:

(23×19)+(25 ⁣:5)+?×3=82 (23\times19)+(25\colon5)+?\times3=82

We solve the exercise in parentheses:

437+5+?×3=82 437+5+?\times3=82

We simplify:

442+?×3=82 442+?\times3=82

We swap sections:

?×3=82442 ?\times3=82-442

?×3=360 ?\times3=-360

We divide by 3

?=120 ?=-120

Answer

?=120 ?=-120

Exercise #1

What is the number that should replace y?

2312×(5)+y ⁣:7+[214]=107 23-12\times(-5)+y\colon7+\lbrack21-4\rbrack=107

Video Solution

Step-by-Step Solution

First, we solve the multiplication exercise:

12×(5)=60 12\times(-5)=-60

and the exercise within brackets:

214=17 21-4=17

We obtain:

23(60)+y:7+17=107 23-(-60)+y:7+17=107

Keep in mind that a negative times a negative becomes a positive:

23+60+y:7+17=107 23+60+y:7+17=107

We simplify and add:

23+60=83 23+60=83

83+17=100 83+17=100

We obtain:

100+y:7=107 100+y:7=107

We move the sections:

y:7=107100 y:7=107-100

y7=7 \frac{y}{7}=7

We multiply by 7:

y=7×7=49 y=7\times7=49

Answer

49

Exercise #2

Daniela goes to the bookshop and buys 4 pens and 9 notebooks for a total of $51.

The price of a pen is twice as much as the price of a notebook.

How much is a pen?

Step-by-Step Solution

We will identify the price of the notebook with x and since the price of the pen is 2 times greater we will mark the price of the pen with 2x

The resulting equation is 4 times the price of a pen plus 9 times the price of a notebook = 51

Now we replace and obtain the following equation:

\( 4\times2x+9\times x=51

According to the rules of the order of arithmetic operations, multiplication and division operations precede addition and subtraction, therefore we will first solve the two multiplication exercises and then add them up:

(4×2x)+(9×x)=51 (4\times2x)+(9\times x)=51

(4×2x)=8x (4\times2x)=8x

(9×x)=9x (9\times x)=9x

8x+9x=17x 8x+9x=17x

Now the obtained equation is: 17x=51 17x=51

We divide both sides by 17 and find x

x=5117=3 x=\frac{51}{17}=3

As we discovered that x is equal to 3, we will place it accordingly and find out the price of a pen:2×x=2×3=6 2\times x=2\times3=6

Answer

6 6

Exercise #3

Mariana has 3 daughters.

The first daughter is 2 times older than the second daughter.

The second daughter is 5 times older than the third daughter.

If we increase the age of the third daughter by 12 years, she will be the same age as the second daughter.

How old is the the first daughter?

Step-by-Step Solution

In the first step, we will try to use variables to change the exercise from verbal to algebraic.

Let's start with the third daughter and define her age as X.

The second daughter, as written, is 5 times older than her, so we will define her age as 5X.

The first daughter is 2 times older than the second daughter, so we will define her age as 2*5X, that is, 10X.

 

Now let's look at the other piece of information: it is known that if we increase the age of the third daughter by 12 years, then she will be the same age as the second sister.

So we will write X+12 (the third daughter plus another 12 years)

=

5X (age of the second daughter)

X + 12 = 5X

 

Once we have an equation, we can solve it. First, we'll move the sections:

5X-X=12

4X=12

We divide by 4:

X=3

 

But this is not the solution!

Remember, we were asked for the age of the first daughter, which is 10X

We replace the X we found:

10*3 = 30

This is the solution!

Answer

30 30

Exercise #4

Find the value of the parameter X:

x+5=8 x+5=8

Video Solution

Answer

3

Exercise #5

Solve for X:

3+x=4 3+x=4

Video Solution

Answer

1

Topics learned in later sections

  1. Solving Equations by Multiplying or Dividing Both Sides by the Same Number
  2. Solving Equations by Simplifying Like Terms
  3. Solving Equations Using the Distributive Property
  4. First-degree equations with one unknown
  5. Solution of an equation