6. a) [tex]x\sqrt{2} = 4 \iff x = \frac{4}{\sqrt{2}} \iff \boxed {x = 2\sqrt{2}}[/tex]
b) [tex]2\sqrt{3} - 3 = x\sqrt{3} \iff \boxed{x = 2 - \sqrt{3}}[/tex]
c) [tex]y\sqrt{2} + 4 = y + 4\sqrt{2}\iff y(\sqrt{2}-1) = 4(\sqrt{2} - 1)\iff \boxed{y = 4}[/tex]
d)[tex]t(3 - \sqrt{2}) = 6 - \sqrt{8}\iff t = 2\cdot\frac{3 - \sqrt{2}}{3 - \sqrt{2}} \iff \boxed{t = 2}[/tex]
e)[tex]z\sqrt{2}-z = 1 +\sqrt{2}\iff z(\sqrt{2} - 1) = \sqrt{2}+1\iff z = \frac{\sqrt{2}+1}{\sqrt{2} - 1} \iff z = \frac{(\sqrt{2})^2 + 2\sqrt{2} + 1}{2-1} \iff \boxed{z = 3 + 2\sqrt{2}}[/tex]
f)[tex]u\sqrt{6} + \sqrt{5} = u\sqrt{3} + \sqrt{10}\iff u(\sqrt{2}\cdot \sqrt{3} - \sqrt{3}) = \sqrt{5}\cdot \sqrt{2} - \sqrt{5} \iff u = \frac{\sqrt{5}(\sqrt{2}-1)}{\sqrt{3}(\sqrt{2} - 1)} \iff u = \frac{\sqrt{5}}{\sqrt{3}} \iff \boxed{u = \frac{\sqrt{15}}{3}}[/tex]
7.a)[tex]3(2x-1) - 5(3x-1) = 1\iff -9x + 2 = 1\iff \boxed{x = \frac{1}{9}}[/tex]
b)[tex]4(x+2) = 2(3x-1) + 5\iff 4x+8 = 6x+3\iff 2x = 5\iff \boxed{x = \frac{5}{2}}[/tex]
c)[tex]3(y+4) - (-3+y)\cdot (-2) = 3\iff 3y + 12 + 2y - 6 = 3\iff 5y = -3\iff \boxed{y = \frac{-3}{5}}[/tex]
d)[tex]6(z+3) - 25 = 5(2z-1)+3z\iff 6z - 7 = 13z - 5\iff -2 = 7z\iff \boxed{z = \frac{-2}{7}}[/tex]
e)[tex]4(t+3)-5(1-2t) = 6t - 1\iff 14t + 7 = 6t - 1\iff 8t = -8\iff \boxed{t = -1}[/tex]
f)[tex]5(x+2)-18 = 2(x - 5) - x\iff 5x - 8 = x - 10 \iff 4x = -2 \iff \boxed{x = \frac{-1}{2}}[/tex]
