Solving linear equations with distributive property

The distributive property states: $a(b+c)=ab+ac$. Every term inside the bracket must be multiplied by the term outside.

Multiply 4 by $x$: $4\times x=4x$.

Multiply 4 by 5: $4\times 5=20$.

Combining both products gives $4x+20$, which is the correct expansion.

We need to correctly expand $4(x+5)$ using the distributive property, meaning 4 must multiply every term inside the bracket.

The option $4x+5$ only multiplied the variable term by 4 but left 5 unchanged. This is the classic mistake of not distributing to every term.

The option $4x+9$ is incorrect — $4+5=9$ suggests someone added instead of multiplied. This is not how distribution works.

The option $x+20$ correctly multiplied 4 by 5 to get 20, but forgot to multiply 4 by $x$, leaving the coefficient of $x$ as 1.

$4x+20$ is correct because $4\times x=4x$ and $4\times 5=20$, so the full expansion is $4x+20$.

The correct expansion of $6(a-3)$ using the distributive property is: $6\times a-6\times 3=6a-18$.

The student wrote $6a-3$. The $6a$ part is correct ($6\times a=6a$), but the constant term should be $6\times 3=18$, not 3.

The student did not multiply 6 by the second term (3). They left the 3 unchanged instead of computing $6\times 3=18$. This means they only distributed to the first term.

The error is that they only multiplied the first term by 6, not both terms. The distributive property requires multiplying every term inside the bracket.

The student wrote $6a-3$ instead of the correct $6a-18$. The $6a$ is right, but the $-3$ is wrong — it should be $-18$.

The original expression has a minus sign inside the bracket, so subtracting is correct here. This is not the error.

If they had multiplied by 3, the answer would be $3a-9$. Since the answer is $6a-3$, this doesn't match the described error.

The student's answer $6a-3$ does include the variable $a$, so this option is not correct.

The error is that they only multiplied the first term ($a$) by 6, but left the second term (3) unchanged. The correct expansion is $6a-18$.