Question

correct your students work
in each of the equalities below your student used exactly one law of algebra.
equality 1
\\(\frac{5}{4+3} = \frac{5}{4} + \frac{5}{3}\\)
which law of algebra did the student use?
the student used the distributive law
the students algebra is incorrect
equality 2
\\(\frac{4+3}{5} = \frac{4}{5} + \frac{3}{5}\\)
which law of algebra did the student use?
the student used the distributive law
the students algebra is correct
equality 3
\\(\frac{4+3}{5} = \frac{5}{4+3}\\)
which law of algebra did the student use?
the student used the associative law

Explanation:

Equality 1

Step1: Recall Distributive Law

The distributive law is \( a(b + c)=ab + ac \), or for division (when distributing the denominator), \( \frac{a + b}{c}=\frac{a}{c}+\frac{b}{c} \) (not \( \frac{a}{b + c}=\frac{a}{b}+\frac{a}{c} \)).

Step2: Analyze Equality 1

The student wrote \( \frac{5}{4 + 3}=\frac{5}{4}+\frac{5}{3} \). This misapplies the distributive law (the distributive law for division distributes the numerator over the denominator, not the denominator over the numerator). So the algebra is incorrect.

Step1: Recall Distributive Law for Division

The distributive law for division over addition in the numerator is \( \frac{a + b}{c}=\frac{a}{c}+\frac{b}{c} \) (where \( c
eq0 \)).

Step2: Analyze Equality 2

Here, \( \frac{4 + 3}{5}=\frac{4}{5}+\frac{3}{5} \) follows \( \frac{a + b}{c}=\frac{a}{c}+\frac{b}{c} \) (with \( a = 4 \), \( b = 3 \), \( c = 5 \)). So the student correctly applied the distributive law.

Step1: Recall Laws of Algebra

The associative law is about grouping (e.g., \( (a + b)+c=a+(b + c) \) for addition, or \( (ab)c=a(bc) \) for multiplication). The equation \( \frac{4 + 3}{5}=\frac{5}{4 + 3} \) is actually an incorrect application of any law (it's not the associative law, and it's not a valid algebraic manipulation). The correct law here would be none, but the student claims associative law. However, let's check: the associative law doesn't apply to this fraction - flipping operation. The left - hand side is \( \frac{7}{5} \), the right - hand side is \( \frac{5}{7} \), which are not equal. So the algebra is incorrect, and the law used is misidentified.

Step2: Correct Law Identification (Optional for Answer)

The operation here is not related to the associative law. The associative law deals with grouping in addition or multiplication, not with inverting the numerator and denominator (which is not a standard algebraic law in this context).

Answer:

The student used the Distributive Law (incorrect application), and the student’s algebra is incorrect.

Equality 2