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.

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 $10$ euros each, so at this moment they have exactly $10$ euros per person.

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

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

## Examples with solutions for Solving Equations by using Addition/ Subtraction

### Exercise #1

$5b+2b-7+14=0$

$b=?$

### Step-by-Step Solution

It's important to remember that when we have regular numbers and unknowns, we cannot add or subtract them directly.

Let's collect like terms:

5b+2b-7+14=0

7b+7 = 0

Let's move terms

7b = -7

Let's divide by 7

b=-1

And that's the solution!

1-

### Exercise #2

Solve for X:

$5x-8=10x+22$

### 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 a$5x$ to the right section and then the 22 to the left side. We obtain the following equation:

$-8-22=10x-5x$

We subtract both sides accordingly and obtain the following equation:

$-30=5x$

We divide both sections by 5 and obtain:

$-6=x$

$-6$

### Exercise #3

What is the missing number?

$23-12\times(-6)+?\colon7=102$

### Step-by-Step Solution

First, we solve the multiplication exercise:

$12\times(-6)=-72$

Now we get:

$23-(-72)+x\colon7=102$

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:

$23+72+x\colon7=102$

We reduce:

$95+x:7=102$

We move the sections:

$x:7=102-95$

$x:7=7$

$\frac{x}{7}=7$

Multiply by 7:

$x=7\times7=49$

49

### Exercise #4

Solve for X:

$\frac{1}{8}x-\frac{2}{3}x+\frac{1}{5}x=1$

### 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:

$(1\times x\times15)-(2\times x\times40)+(1\times x\times24)=1\times120$

We multiply the exercises in parentheses accordingly:

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

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

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

$-65x+24x=120$

$-41x=120$

We reduce both sides by $-41$

$\frac{-41x}{-41}=\frac{120}{-41}$

We find that x is equal$x=-\frac{120}{41}$

$-\frac{120}{41}$

### Exercise #5

Fill in the missing number:

$?\colon(-6)+\lbrack-2(94-73)\rbrack=-153$

### Step-by-Step Solution

We begin by solving the innermost parentheses:

$94-73=21$

We obtain the following calculation:

$x\colon(-6)+\lbrack-2(21)\rbrack=-153$

Next we focus on the parentheses and multiply:

$-2\times21=-42$

We obtain the following calculation:

$x\colon(-6)+(-42)=-153$

Let's consider the multiplication between minus and plus:

$x\colon(-6)-42=-153$

We swap sections:

$x\colon(-6)=-153+42$

$x\colon(-6)=-111$

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

Lastly we multiply by minus 6:

$x=-111\times-6=666$

$x=666$

### Exercise #6

Solve the equation and find Y:

$20\times y+8\times2-7=14$

### Step-by-Step Solution

We begin by placing parentheses around the two multiplication exercises:

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

We then solve the exercises within the parentheses:

$20y+16-7=14$

We simplify:

$20y+9=14$

We move the sections:

$20y=14-9$

$20y=5$

We divide by 20:

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

$y=\frac{5}{5\times4}$

We simplify:

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

$\frac{1}{4}$

### Exercise #7

Fill in the missing number:

$?\colon4+8\times3-7=26$

### Step-by-Step Solution

First, we place the multiplication exercise inside of parentheses:

$x\colon4+(8\times3)-7=26$

Next we solve the multiplication exercise:

$x\colon4+24-7=26$

We then place the subtraction exercise in parentheses:

$x\colon4+(24-7)=26$

We solve the subtraction exercise:

$x\colon4+17=26$

We swap sections:

$x\colon4=26-17$

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

Multiply by 4

$x=4\times9=36$

$?=36$

### Exercise #8

$92-x\times2-24\colon4=64$Calculate X.

### Step-by-Step Solution

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

$92-(x\times2)-(24\colon4)=64$

$92-2x-6=64$

Reduce:

$86-2x=64$

Move the sides:

$-2x=64-86$

$-2x=-22$

Divide by negative 2:

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

$x=11$

11

### Exercise #9

How much is x equal to?

$-25-42\colon x+18\times2\colon4=-23$

### Step-by-Step Solution

We begin by placing the multiplication exercise inside of parentheses:

$-25-42\colon x+(18\times2)\colon4=-23$

$-25-42\colon x+36\colon4=-23$

We will then place the division exercise inside of parentheses:

$-25-42\colon x+(36\colon4)=-23$

$-25-42\colon x+9=-23$

Next we rearrange the exercise in order to simplify it:

$-25+9-42\colon x=-23$

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

We then solve the exercise inside of the parenthesis and obtain the following:

$-16-42\colon x=-23$

We rearrange the fractions and obtain the following:

$-42\colon x=-23+16$

$-42\colon x=-7$

We multiply by x and obtain the following:

$-42=-7x$

Lastly we divide by negative 7:

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

6

### Exercise #10

What is the missing number in the equation?

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

### Step-by-Step Solution

We place the multiplication exercise inside of the parenthesis:

$(5\times17)-112-\frac{x}{14}=-27.5$

We solve the multiplication exercise and then subtract:

$85-112-\frac{x}{14}=-27.5$

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

We swap sections:

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

$-\frac{x}{14}=-\frac{1}{2}$

We multiply by 14:

$-x=-\frac{1}{2}\times14$

$-x=-7$

We multiply by -1

$x=7$

$7$

### Exercise #11

Calculate the missing number in the equation:

$(45-22)\times19+25\colon5+?\times3=82$

### Step-by-Step Solution

We solve the exercise in parentheses:

$23\times19+25\colon5+?\times3=82$

We place the multiplication and division exercises inside parentheses:

$(23\times19)+(25\colon5)+?\times3=82$

We solve the exercise in parentheses:

$437+5+?\times3=82$

We simplify:

$442+?\times3=82$

We swap sections:

$?\times3=82-442$

$?\times3=-360$

We divide by 3

$?=-120$

$?=-120$

### Exercise #12

What is the number that should replace y?

$23-12\times(-5)+y\colon7+\lbrack21-4\rbrack=107$

### Step-by-Step Solution

We begin by solving the multiplication exercise:

$12\times(-5)=-60$

and subsequently the exercises within brackets:

$21-4=17$

We obtain the following:

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

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

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

Next we simplify and add:

$23+60=83$

$83+17=100$

We obtain the following calculation:

$100+y:7=107$

We then rearrange the sections:

$y:7=107-100$

$\frac{y}{7}=7$

Lastly we multiply by 7:

$y=7\times7=49$

49

### Exercise #13

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:

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

$(4\times2x)=8x$

$(9\times x)=9x$

$8x+9x=17x$

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

We divide both sides by 17 and find x

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

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

$6$

### Exercise #14

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!

$30$

### Exercise #15

$x+7=14$