From Arithmetic to Algebra: Practice

Distribute, then combine the like terms.

$2(x + 3) + 4x = 2x + 6 + 4x = 6x + 6$

The leading minus sign multiplies each term inside the parentheses by $-1$.

$-(x - 4) = -x + 4 = 4 - x$

Adding three copies of $n$ is the meaning of $3n$, so the two sides match no matter what $n$ is.

$n + n + n = 3n$

This is an identity, true for every $n$.

"Twice the sum of a number and $3$" is $2(n + 3)$, and "is the same as three times the number" sets it equal to $3n$.

$2(n + 3) = 3n$

Factoring starts from the expanded sum and pulls out a common factor, so the sum is on the left and the product on the right.

$6x + 9 = 3(2x + 3)$

The reverse direction, $3(2x + 3) = 6x + 9$, is the distributive law used to expand a product instead.

Substitute, resolve the inner parentheses first, then subtract.

$5 - (6 - 2) = 5 - 4 = 1$

The parentheses group $n - 2$, so it is evaluated before the outer subtraction.

Distribute both products, then combine the like terms.

$3(x + 2) + 2(x + 1) = 3x + 6 + 2x + 2 = 5x + 8$

The notation $2xy$ means $2 \cdot x \cdot y$. Substitute and multiply, tracking the sign.

$2(-3)(4) = -24$

Every term shares the factor $a$, and the last term $a$ is $a \cdot 1$.

$ab + ac + a = a(b + c + 1)$

Check by distributing: $a \cdot b + a \cdot c + a \cdot 1$.

Substitute $a = 5$ into $a \cdot b = 0$ to get a one-step equation in $b$, then divide both sides by $5$.

$5b = 0 \;\Rightarrow\; b = \frac{0}{5} = 0$

So $b = 0$. Check: $5 \cdot 0 = 0$, as required.

"Three less than twice a number" is $2n - 3$ (start from $2n$, then take $3$ away), and "the number plus four" is $n + 4$.

$2n - 3 = n + 4$

The distributive law multiplies the outside factor by each term in the sum.

$4(x + y) = 4x + 4y$

The others misapply it: distributing must reach every term, and $4(xy) = 4xy$, not $4x \cdot 4y$.