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

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.

Detailed explanation

Start practice

Test yourself on solution of an equation by adding/subtracting two sides!

$$ x+7=14 $$

Practice more now

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

Example 1

$X+5+2=3$

If we are asked what the value of the expression $X+5$ is, we can leave it on the left side of the equation if we subtract the number $2$ from both sides.

$X+5+2=3$ / $-2$

$X+5=1$

Here we see that the expression $X+5$ is equivalent to $1$ .

Example 2

$X+7-4=10$

If we are asked what the value of the expression $X+7$ is, we can leave it on the left side of the equation if we add the number $4$ to both sides of the equation.

$X+7-4=10$ / $+4$

$X+7=14$

Here we see that the expression $X+7$ is equivalent to $14$ .

Examples and exercises with solutions for solving equations by adding or subtracting the same number from both sides

examples.example_title

Find the value of the parameter X

$5x-8=10x+22$

examples.explanation_title

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:

$-6=x$

examples.solution_title

$-6$

examples.example_title

Find the value of parameter X

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

examples.explanation_title

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

examples.solution_title

$-\frac{120}{41}$

examples.example_title

When Daniela went to the bookstore she bought 4 pens and 9 notebooks for $51.

It is known that the price of the pen is twice as much as the price of the notebook.

What is the price of the pen?

examples.explanation_title

We will identify the price of the notebook with x and as the price of the pen is twice as much 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:

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

$(4\times2x)=8x$

$(9\times x)=9x$

$8x+9x=17x$

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 equals 3, we will place it accordingly and find out the price of a pen: $2\times x=2\times3=6$

examples.solution_title

$6$

examples.example_title

Mariana has 3 daughters.

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

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

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

Find the age of the first sister.

examples.explanation_title

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, 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