to assume the responsibility for paying another person's debt
「借金の肩代わり」は to take over someone's debt や to assume the responsibility for paying another person's debt のように言えます。
take over「引き継ぐ・取って代わる・肩代わりする」
debt「借金・負債」
assume「（責任などを）とる・引き受ける」
responsibility 「責任」
My brother lost his job after he had just bought a new car. He couldn't make his car payments so I had to assume the responsibility for his payments since I had co-signed his loan.
「弟は新しい車を買ってすぐ職を失った。車の支払いができなかったので、車のローンの保証人となっていた私が支払いを肩代わりしなければならなかった。」
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カナダ 
