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

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

Solve for X:

$1.5-x=\text{2}.8$

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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

Solve the equation and find Y:

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

examples.explanation_title

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

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

We 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:

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

examples.solution_title

examples.example_title

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

examples.explanation_title

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:

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

$x=11$

examples.solution_title

examples.example_title

Solve for 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

What is the missing number?

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

examples.explanation_title

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:

$x=7\times7=49$

examples.solution_title

examples.example_title

Solve for 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

$-\frac{120}{41}$

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Test your knowledge

Question 1

Solve for X:

$$ x+3=5 $$

Question 2

Solve for X:

$$ 3-x=1 $$

Question 3

Find the value of the parameter X:

$$ x+5=8 $$

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