You used the distributive property to factor sums of numbers like $36 + 8 = 4(9 + 2)$. It works the same way with variables, and it lets you rewrite an expression without changing its value.
Idea clave
Equivalent expressions are equal for every value of the variable. The distributive property, $a(b + c) = ab + ac$, makes them: $3(x + 2) = 3x + 6$. So does combining like terms: $y + y + y = 3y$.
Por qué funciona
Picture a rectangle 3 units tall and $x + 2$ units long. Split it into a 3 by $x$ part (area $3x$) and a 3 by 2 part (area 6). The whole area is $3(x + 2)$; the parts add to $3x + 6$. Since it is the same rectangle, the two expressions are equal whatever length $x$ is. Combining like terms is the same property: $4a + 3a = (4 + 3)a = 7a$.
Ejemplo resuelto: Expand
Write $3(x + 2)$ without parentheses.
- The 3 multiplies each term inside: $3 \times x = 3x$ and $3 \times 2 = 6$.
Respuesta: $3x + 6$
Ejemplo resuelto: Factor
Factor $12 + 18x$.
- The greatest common factor of 12 and 18 is 6.
- $12 = 6 \times 2$ and $18x = 6 \times 3x$.
- So $12 + 18x = 6(2 + 3x)$.
Respuesta: $6(2 + 3x)$
Ejemplo resuelto: Combine like terms
Simplify $4a + 2 + 3a$.
- $4a$ and $3a$ are like terms: both count $a$’s.
- $4a + 3a = 7a$. The 2 has no $a$, so it stays.
Respuesta: $7a + 2$
Error común
Multiplying only the first term: $3(x + 2) = 3x + 2$. Quick check: try $x = 1$. $3(1 + 2) = 9$, but $3(1) + 2 = 5$. They are not equal, so something was left out. The 3 must multiply the 2 as well.