query: For any numbers $a, b,$ and $c,$ $a + b = a + c$ if and only if $b = c$
positive_passages.docid: 10508
positive_passages.text: Algebra: What allows us to do the same thing to both sides of an equation?
negative_passages.docid: 124742
negative_passages.text: P(A|C)=P(A|B)*P(B|C)?
negative_passages.docid: 11868
negative_passages.text: Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$
negative_passages.docid: 68743
negative_passages.text: If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ then $\frac{1}{a^5}+\frac{1}{b^5}+\frac{1}{c^5}=\frac{1}{a^5+b^5+c^5}.$
negative_passages.docid: 87013
negative_passages.text: If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$
negative_passages.docid: 26861
negative_passages.text: $a+b=c \times d$ and $a\times b = c + d$
negative_passages.docid: 80095
negative_passages.text: Show $|a|+|b|+|c|+|a+b+c| \geq  |a+b|+|b+c|+|c+a|$ for complex $a$, $b$, $c$
negative_passages.docid: 104068
negative_passages.text: Prove that $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$
negative_passages.docid: 53660
negative_passages.text: Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $
negative_passages.docid: 151607
negative_passages.text: Show that  $A\Delta B = C$ if and only if $A = B\Delta C$
negative_passages.docid: 12954
negative_passages.text: Prove if $ac>bc$ and $c>0$, then $a>b$
negative_passages.docid: 153053
negative_passages.text: Find numbers $a, b, c$ given that $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$
negative_passages.docid: 11168
negative_passages.text: Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
negative_passages.docid: 70311
negative_passages.text: Trying to prove (A ∩ B) − C = (A − C) ∩ (B − C)
negative_passages.docid: 19909
negative_passages.text: if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$
negative_passages.docid: 115929
negative_passages.text: In a triangle $\angle A = 2\angle B$ iff $a^2 = b(b+c)$
negative_passages.docid: 1713
negative_passages.text: number theory $\gcd(a,bc)=\gcd(a,c)$