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Proof verification:

Let $(X,A)$ be a pair of topological spaces and suppose that $A$ is path-connected. If $j:(X,\emptyset)\to (X,A)$, $x\mapsto j(x)=x$ then $j_*:H_1(X)\to H_1(X,A)$ is surjective.

Let $T+S_1(X)$ be an element of $Z_1(X,A)$. If we choose $T'=T$ then: $$j_*[T]=[T+S_1(X)]$$ So, it is clear that $j_*$ is surjective

What is wrong with my reasoning? (I did not use that $A$ is path connected)

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    What are $T$ and $S_1(X)$? – 2017-01-23
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    Just answering with a guess on the terminologies, I think your $T^\prime$ is not necessarily in $Z_1(X)$. – 2017-01-23

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It's much easier to approach this using the long exact sequence of a pair $(X,A)$ (for reduced homology): $$\cdots\xrightarrow{\ \ \ }\tilde{H}_1(A)\xrightarrow{\ i \ }\tilde{H}_1(X)\xrightarrow{\ j \ }\tilde{H}_1(X,A)\xrightarrow{\ \partial \ }\tilde{H}_0(A)\xrightarrow{\ \ \ }\cdots$$ Note that $\tilde{H}_1(X) = H_1(X)$ and $\tilde{H}_1(X,A)=H_1(X,A)$. What does $A$ being path connected imply here?