Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+20& = & 108x+180 \\\Leftrightarrow & 45x \color{red}{+20} \color{blue}{-20} \color{blue}{-108x} & = & \color{red}{108x} +180 \color{blue}{-108x} \color{blue}{-20} \\\Leftrightarrow & -63x& = & 160 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{160}{-63} \\\Leftrightarrow & x = \frac{-160}{63} & & \\ & V = \left\{ \frac{-160}{63} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-18& = & 39x+390 \\\Leftrightarrow & 26x \color{red}{-18} \color{blue}{+18} \color{blue}{-39x} & = & \color{red}{39x} +390 \color{blue}{-39x} \color{blue}{+18} \\\Leftrightarrow & -13x& = & 408 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{408}{-13} \\\Leftrightarrow & x = \frac{-408}{13} & & \\ & V = \left\{ \frac{-408}{13} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{2}{3}x+6 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{104}{ \color{blue}{156} }x+\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & 104x+936 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{-104x} & = & \color{red}{104x} +936 \color{blue}{-104x} \color{blue}{-48} \\\Leftrightarrow & -65x& = & 888 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{888}{-65} \\\Leftrightarrow & x = \frac{-888}{65} & & \\ & V = \left\{ \frac{-888}{65} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 28x-84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-28x} & = & \color{red}{28x} -84 \color{blue}{-28x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -60 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-60}{-7} \\\Leftrightarrow & x = \frac{60}{7} & & \\ & V = \left\{ \frac{60}{7} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 6 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-25& = & -12x+240 \\\Leftrightarrow & 15x \color{red}{-25} \color{blue}{+25} \color{blue}{+12x} & = & \color{red}{-12x} +240 \color{blue}{+12x} \color{blue}{+25} \\\Leftrightarrow & 27x& = & 265 \\\Leftrightarrow & \frac{27x}{ \color{red}{27} }& = & \frac{265}{27} \\\Leftrightarrow & x = \frac{265}{27} & & \\ & V = \left\{ \frac{265}{27} \right\} & \\\end{align}\)
6. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{-5}{6}x-8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 70 }{ \color{blue}{126} })& = & (\frac{-105}{ \color{blue}{126} }x-\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+70& = & -105x-1008 \\\Leftrightarrow & 18x \color{red}{+70} \color{blue}{-70} \color{blue}{+105x} & = & \color{red}{-105x} -1008 \color{blue}{+105x} \color{blue}{-70} \\\Leftrightarrow & 123x& = & -1078 \\\Leftrightarrow & \frac{123x}{ \color{red}{123} }& = & \frac{-1078}{123} \\\Leftrightarrow & x = \frac{-1078}{123} & & \\ & V = \left\{ \frac{-1078}{123} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 3, 14 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{14}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{70x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 70x-75& = & -168x+210 \\\Leftrightarrow & 70x \color{red}{-75} \color{blue}{+75} \color{blue}{+168x} & = & \color{red}{-168x} +210 \color{blue}{+168x} \color{blue}{+75} \\\Leftrightarrow & 238x& = & 285 \\\Leftrightarrow & \frac{238x}{ \color{red}{238} }& = & \frac{285}{238} \\\Leftrightarrow & x = \frac{285}{238} & & \\ & V = \left\{ \frac{285}{238} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x-\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & -130x-624 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{+130x} & = & \color{red}{-130x} -624 \color{blue}{+130x} \color{blue}{+18} \\\Leftrightarrow & 169x& = & -606 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-606}{169} \\\Leftrightarrow & x = \frac{-606}{169} & & \\ & V = \left\{ \frac{-606}{169} \right\} & \\\end{align}\)
9. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 28 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{154}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-28& = & 77x+154 \\\Leftrightarrow & 22x \color{red}{-28} \color{blue}{+28} \color{blue}{-77x} & = & \color{red}{77x} +154 \color{blue}{-77x} \color{blue}{+28} \\\Leftrightarrow & -55x& = & 182 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{182}{-55} \\\Leftrightarrow & x = \frac{-182}{55} & & \\ & V = \left\{ \frac{-182}{55} \right\} & \\\end{align}\)
10. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{90}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-8& = & 9x-90 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-9x} & = & \color{red}{9x} -90 \color{blue}{-9x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & -82 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-82}{-3} \\\Leftrightarrow & x = \frac{82}{3} & & \\ & V = \left\{ \frac{82}{3} \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x+\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -264x+1320 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+264x} & = & \color{red}{-264x} +1320 \color{blue}{+264x} \color{blue}{+150} \\\Leftrightarrow & 319x& = & 1470 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{1470}{319} \\\Leftrightarrow & x = \frac{1470}{319} & & \\ & V = \left\{ \frac{1470}{319} \right\} & \\\end{align}\)
12. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{5}{4}x+4 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }- \frac{ 84 }{ \color{blue}{308} })& = & (\frac{385}{ \color{blue}{308} }x+\frac{1232}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x-84& = & 385x+1232 \\\Leftrightarrow & 44x \color{red}{-84} \color{blue}{+84} \color{blue}{-385x} & = & \color{red}{385x} +1232 \color{blue}{-385x} \color{blue}{+84} \\\Leftrightarrow & -341x& = & 1316 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{1316}{-341} \\\Leftrightarrow & x = \frac{-1316}{341} & & \\ & V = \left\{ \frac{-1316}{341} \right\} & \\\end{align}\)