In Grade 6 you saw that expressions can look different and still be equal, like $y + y + y$ and $3y$. Combining like terms is the everyday way to make an expression shorter without changing its value. The coefficients can be negative, fractions or decimals, so you will use signed-number addition too.
Idea clave
Like terms have exactly the same variable part, such as $3x$ and $-5x$, or $4$ and $-7$. Combine like terms by adding their coefficients: $3x + 5x = 8x$. Unlike terms, such as $3x$ and $2$, stay separate.
Por qué funciona
$3x + 5x$ is 3 groups of $x$ and 5 more groups of $x$: 8 groups in all. That is the distributive property, $3x + 5x = (3 + 5)x$. But $3x + 2$ is 3 groups of $x$ and 2 ones. Groups of $x$ and ones are different things, like cups and spoons, so they cannot be counted together.
Worked examples
Ejemplo resuelto: Two kinds of terms
Simplify $3x + 5 - 7x + 2$.
- Group the $x$-terms and the constants: $(3x - 7x) + (5 + 2)$.
- $3 + (-7) = -4$, so the $x$-terms make $-4x$. The constants make $7$.
Respuesta: $-4x + 7$
Ejemplo resuelto: Two variables
Simplify $4a - 6b + a + 2b - 3$.
- $a$-terms: $4a + a = 5a$ (remember $a$ means $1a$).
- $b$-terms: $-6b + 2b = -4b$. Constant: $-3$.
Respuesta: $5a - 4b - 3$
Ejemplo resuelto: Fraction coefficients
Simplify $\frac{1}{2}y + \frac{3}{4} - \frac{1}{4}y + 1$.
- $y$-terms: $\frac{1}{2} - \frac{1}{4} = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}$, so $\frac{1}{4}y$.
- Constants: $\frac{3}{4} + 1 = \frac{7}{4}$.
Respuesta: $\frac{1}{4}y + \frac{7}{4}$
Error común
Two classic slips: $3x + 2 = 5x$ (adding unlike terms), and $x + x = x^2$ (adding is not multiplying: $x + x = 2x$, while $x \cdot x = x^2$). Test with a number: if $x = 10$, then $3x + 2 = 32$ but $5x = 50$, so they are not equal.