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 solving equations by using addition/ subtraction!

Find the value of the parameter X

$$ -8-x=5 $$

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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

Exercise #1

Solve for A:

$a-5=10$

Step-by-Step Solution

To solve for $a$ , we need to isolate it on one side of the equation. Starting with:

$a-5=10$

Add $5$ to both sides to get:

$a-5+5=10+5$

This simplifies to:

$a=15$

Therefore, the solution is $a = 15$ .

Answer

$15$

Exercise #2

Solve for B:

$b+6=14$

Step-by-Step Solution

To solve for $b$ , we need to isolate it on one side of the equation. Starting with:

$b+6=14$

Subtract $6$ from both sides to get:

$b+6-6=14-6$

This simplifies to:

$b=8$

Therefore, the solution is $b = 8$ .

Answer

$8$

Exercise #3

Solve for X:

$x + 3 = 7$

Step-by-Step Solution

To solve for $x$ , start by isolating $x$ on one side of the equation:
Subtract $3$ from both sides:
$x + 3 - 3 = 7 - 3$ simplifies to
$x = 4$ .

Answer

Exercise #4

Solve for X:

$x - 5 = -10$

Step-by-Step Solution

To solve the equation $x - 5 = -10$ , we need to isolate $x$ .

Step 1: Add 5 to both sides of the equation to cancel out the -5 on the left side.
$x - 5 + 5 = -10 + 5$
Step 2: Simplify both sides.
$x = -5$
Thus, the solution is $x = -5$ .