Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{2}{13}=\frac{1}{3}x+3\)
2. \(\frac{x}{7}-\frac{5}{14}=\frac{1}{2}x-4\)
3. \(\frac{x}{6}+\frac{4}{15}=\frac{-4}{5}x-6\)
4. \(\frac{x}{4}+\frac{4}{9}=\frac{-8}{3}x+5\)
5. \(\frac{x}{7}-\frac{3}{13}=\frac{5}{6}x+2\)
6. \(\frac{x}{2}-\frac{5}{11}=\frac{-8}{3}x+5\)
7. \(\frac{x}{6}+\frac{4}{9}=\frac{6}{5}x-2\)
8. \(\frac{x}{5}+\frac{4}{11}=\frac{5}{6}x+6\)
9. \(\frac{x}{7}-\frac{2}{7}=\frac{1}{6}x+5\)
10. \(\frac{x}{6}-\frac{2}{7}=\frac{-4}{5}x+8\)
11. \(\frac{x}{5}+\frac{4}{7}=\frac{7}{4}x+3\)
12. \(\frac{x}{3}-\frac{4}{7}=\frac{1}{2}x+6\)

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Reeks met breuken waarbij we ervoor kiezen om de breuken weg te werken. Neem je ZRM!

Verbetersleutel

1. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x+\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & 26x+234 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{-26x} & = & \color{red}{26x} +234 \color{blue}{-26x} \color{blue}{-12} \\\Leftrightarrow & 13x& = & 222 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{222}{13} \\\Leftrightarrow & x = \frac{222}{13} & & \\ & V = \left\{ \frac{222}{13} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{14}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 5 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{56}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-5& = & 7x-56 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-7x} & = & \color{red}{7x} -56 \color{blue}{-7x} \color{blue}{+5} \\\Leftrightarrow & -5x& = & -51 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-51}{-5} \\\Leftrightarrow & x = \frac{51}{5} & & \\ & V = \left\{ \frac{51}{5} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & -24x-180 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{+24x} & = & \color{red}{-24x} -180 \color{blue}{+24x} \color{blue}{-8} \\\Leftrightarrow & 29x& = & -188 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-188}{29} \\\Leftrightarrow & x = \frac{-188}{29} & & \\ & V = \left\{ \frac{-188}{29} \right\} & \\\end{align}\)
4. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-96}{ \color{blue}{36} }x+\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & -96x+180 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{+96x} & = & \color{red}{-96x} +180 \color{blue}{+96x} \color{blue}{-16} \\\Leftrightarrow & 105x& = & 164 \\\Leftrightarrow & \frac{105x}{ \color{red}{105} }& = & \frac{164}{105} \\\Leftrightarrow & x = \frac{164}{105} & & \\ & V = \left\{ \frac{164}{105} \right\} & \\\end{align}\)
5. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 126 }{ \color{blue}{546} })& = & (\frac{455}{ \color{blue}{546} }x+\frac{1092}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-126& = & 455x+1092 \\\Leftrightarrow & 78x \color{red}{-126} \color{blue}{+126} \color{blue}{-455x} & = & \color{red}{455x} +1092 \color{blue}{-455x} \color{blue}{+126} \\\Leftrightarrow & -377x& = & 1218 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{1218}{-377} \\\Leftrightarrow & x = \frac{-42}{13} & & \\ & V = \left\{ \frac{-42}{13} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x+\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & -176x+330 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{+176x} & = & \color{red}{-176x} +330 \color{blue}{+176x} \color{blue}{+30} \\\Leftrightarrow & 209x& = & 360 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{360}{209} \\\Leftrightarrow & x = \frac{360}{209} & & \\ & V = \left\{ \frac{360}{209} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x-\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 108x-180 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-108x} & = & \color{red}{108x} -180 \color{blue}{-108x} \color{blue}{-40} \\\Leftrightarrow & -93x& = & -220 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{-220}{-93} \\\Leftrightarrow & x = \frac{220}{93} & & \\ & V = \left\{ \frac{220}{93} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{5}{6}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{275}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & 275x+1980 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{-275x} & = & \color{red}{275x} +1980 \color{blue}{-275x} \color{blue}{-120} \\\Leftrightarrow & -209x& = & 1860 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{1860}{-209} \\\Leftrightarrow & x = \frac{-1860}{209} & & \\ & V = \left\{ \frac{-1860}{209} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-12& = & 7x+210 \\\Leftrightarrow & 6x \color{red}{-12} \color{blue}{+12} \color{blue}{-7x} & = & \color{red}{7x} +210 \color{blue}{-7x} \color{blue}{+12} \\\Leftrightarrow & -x& = & 222 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{222}{-1} \\\Leftrightarrow & x = -222 & & \\ & V = \left\{ -222 \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x+1680 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} +1680 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & 1740 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{1740}{203} \\\Leftrightarrow & x = \frac{60}{7} & & \\ & V = \left\{ \frac{60}{7} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{7}{4}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{245}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+80& = & 245x+420 \\\Leftrightarrow & 28x \color{red}{+80} \color{blue}{-80} \color{blue}{-245x} & = & \color{red}{245x} +420 \color{blue}{-245x} \color{blue}{-80} \\\Leftrightarrow & -217x& = & 340 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{340}{-217} \\\Leftrightarrow & x = \frac{-340}{217} & & \\ & V = \left\{ \frac{-340}{217} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x+252 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} +252 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & 276 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{276}{-7} \\\Leftrightarrow & x = \frac{-276}{7} & & \\ & V = \left\{ \frac{-276}{7} \right\} & \\\end{align}\)