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We improve on Jarn\'{\i}k's inequality between uniform Diophantine exponent$\alpha $ and ordinary Diophantine exponent $\beta$ for a system of $ n\ge 2$real linear forms in two integer variables. Jarn\'{\i}k (1949, 1954) provedthat $\beta \ge \alpha (\alpha -1)$. In the present paper we give a betterbound in the case $\alpha >1$. We prove that \beta \ge1/2(\alpha^2-\alpha+1+\sqrt{(\alpha^2-\alpha+1)^2 +4\alpha^2(\alpha-1)}) if1\le \alpha \le 2 1/2(\alpha^2-1+\sqrt{(\alpha^2-1)^2+4\alpha (\alpha-1)}) if\alpha \ge 2

Diophantine exponents for systems of linear forms in two variables