Adding or subtracting a constant to each side of an equation will not change the equality.

For example, for the equation

$$x+2=5,$$

we can add 3 to both sides to get

$$x+2+3=5+3,$$

which simplifies to

$$x+5=8$$

We can also subtract 2 from both sides to get

$$x+2-2=5-2,$$

which simplifies to

$$x=3.$$

2. Multiplication or Division

Multiplying or dividing both sides of an equation by a nonzero constant will not change the equality.

For example, for the equation

$$3x=15,$$

we can divide both sides by 3 to get

$$\frac{3x}{3}=\frac{15}{3},$$

which simplifies to

$$x=5.$$

We can also multiply both sides by 2 to get

$$3x\cdot2=15\cdot2,$$

which simplifies to

$$6x=30$$

3. Factoring

Factoring is a process of writing an expression as a product of simpler expressions.

For example, for the equation

$$x^2+2x-3=0,$$

we can factor as follows:

$$(x+3)(x-1)=0$$

Setting each factor equal to zero, we get

$$x+3=0 \quad \text{or} \quad x-1=0$$

Solving each equation, we get

$$x=-3 \quad \text{or} \quad x=1$$

4. Completing the square

Completing the square is a process of transforming a quadratic equation into a perfect square.

For example, for the equation

$$x^2-4x-5=0,$$

we can complete the square as follows:

$$x^2-4x+4-4-5=0$$

$$(x-2)^2-9=0$$

Adding 9 to both sides, we get

$$(x-2)^2=9$$

Taking the square root of both sides, we get

$$x-2=\pm3$$

Solving each equation, we get

$$x=2+3=5 \quad \text{or} \quad x=2-3=-1$$

5. Substitution

Substitution is a process of replacing one expression with another equivalent expression.

For example, for the equation

$$y=3x+2$$

we can substitute \(y\) with \(x+5\):

$$x+5=3x+2$$

Solving for \(x\):

$$x-3x=-5+2$$

$$-2x=-3$$

$$x=\frac{3}{2}$$