Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}-\frac{5}{16}=\frac{1}{2}x-2\)
2. \(\frac{x}{7}+\frac{5}{9}=\frac{5}{4}x-5\)
3. \(\frac{x}{4}+\frac{4}{11}=\frac{1}{3}x+1\)
4. \(\frac{x}{6}+\frac{5}{14}=\frac{6}{5}x+5\)
5. \(\frac{x}{2}+\frac{4}{11}=\frac{-5}{3}x-4\)
6. \(\frac{x}{3}-\frac{5}{12}=\frac{7}{4}x+7\)
7. \(\frac{x}{5}+\frac{5}{6}=\frac{5}{2}x-7\)
8. \(\frac{x}{2}+\frac{2}{11}=\frac{-7}{5}x-5\)
9. \(\frac{x}{6}+\frac{2}{13}=\frac{7}{5}x-4\)
10. \(\frac{x}{2}+\frac{3}{13}=\frac{-2}{3}x-4\)
11. \(\frac{x}{2}-\frac{2}{9}=\frac{-2}{5}x+8\)
12. \(\frac{x}{2}-\frac{5}{7}=\frac{4}{3}x-3\)

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

Verbetersleutel

1. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x-\frac{224}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & 56x-224 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{-56x} & = & \color{red}{56x} -224 \color{blue}{-56x} \color{blue}{+35} \\\Leftrightarrow & -40x& = & -189 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-189}{-40} \\\Leftrightarrow & x = \frac{189}{40} & & \\ & V = \left\{ \frac{189}{40} \right\} & \\\end{align}\)
2. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{5}{4}x-5 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }+ \frac{ 140 }{ \color{blue}{252} })& = & (\frac{315}{ \color{blue}{252} }x-\frac{1260}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x+140& = & 315x-1260 \\\Leftrightarrow & 36x \color{red}{+140} \color{blue}{-140} \color{blue}{-315x} & = & \color{red}{315x} -1260 \color{blue}{-315x} \color{blue}{-140} \\\Leftrightarrow & -279x& = & -1400 \\\Leftrightarrow & \frac{-279x}{ \color{red}{-279} }& = & \frac{-1400}{-279} \\\Leftrightarrow & x = \frac{1400}{279} & & \\ & V = \left\{ \frac{1400}{279} \right\} & \\\end{align}\)
3. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x+\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & 44x+132 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{-44x} & = & \color{red}{44x} +132 \color{blue}{-44x} \color{blue}{-48} \\\Leftrightarrow & -11x& = & 84 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{84}{-11} \\\Leftrightarrow & x = \frac{-84}{11} & & \\ & V = \left\{ \frac{-84}{11} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & 252x+1050 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{-252x} & = & \color{red}{252x} +1050 \color{blue}{-252x} \color{blue}{-75} \\\Leftrightarrow & -217x& = & 975 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{975}{-217} \\\Leftrightarrow & x = \frac{-975}{217} & & \\ & V = \left\{ \frac{-975}{217} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-110}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+24& = & -110x-264 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+110x} & = & \color{red}{-110x} -264 \color{blue}{+110x} \color{blue}{-24} \\\Leftrightarrow & 143x& = & -288 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-288}{143} \\\Leftrightarrow & x = \frac{-288}{143} & & \\ & V = \left\{ \frac{-288}{143} \right\} & \\\end{align}\)
6. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{12}& = & \frac{7}{4}x+7 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{21}{ \color{blue}{12} }x+\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-5& = & 21x+84 \\\Leftrightarrow & 4x \color{red}{-5} \color{blue}{+5} \color{blue}{-21x} & = & \color{red}{21x} +84 \color{blue}{-21x} \color{blue}{+5} \\\Leftrightarrow & -17x& = & 89 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{89}{-17} \\\Leftrightarrow & x = \frac{-89}{17} & & \\ & V = \left\{ \frac{-89}{17} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{75}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 75x-210 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-75x} & = & \color{red}{75x} -210 \color{blue}{-75x} \color{blue}{-25} \\\Leftrightarrow & -69x& = & -235 \\\Leftrightarrow & \frac{-69x}{ \color{red}{-69} }& = & \frac{-235}{-69} \\\Leftrightarrow & x = \frac{235}{69} & & \\ & V = \left\{ \frac{235}{69} \right\} & \\\end{align}\)
8. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{-154}{ \color{blue}{110} }x-\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+20& = & -154x-550 \\\Leftrightarrow & 55x \color{red}{+20} \color{blue}{-20} \color{blue}{+154x} & = & \color{red}{-154x} -550 \color{blue}{+154x} \color{blue}{-20} \\\Leftrightarrow & 209x& = & -570 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-570}{209} \\\Leftrightarrow & x = \frac{-30}{11} & & \\ & V = \left\{ \frac{-30}{11} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{7}{5}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{546}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & 546x-1560 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-546x} & = & \color{red}{546x} -1560 \color{blue}{-546x} \color{blue}{-60} \\\Leftrightarrow & -481x& = & -1620 \\\Leftrightarrow & \frac{-481x}{ \color{red}{-481} }& = & \frac{-1620}{-481} \\\Leftrightarrow & x = \frac{1620}{481} & & \\ & V = \left\{ \frac{1620}{481} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & -52x-312 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{+52x} & = & \color{red}{-52x} -312 \color{blue}{+52x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & -330 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-330}{91} \\\Leftrightarrow & x = \frac{-330}{91} & & \\ & V = \left\{ \frac{-330}{91} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-20& = & -36x+720 \\\Leftrightarrow & 45x \color{red}{-20} \color{blue}{+20} \color{blue}{+36x} & = & \color{red}{-36x} +720 \color{blue}{+36x} \color{blue}{+20} \\\Leftrightarrow & 81x& = & 740 \\\Leftrightarrow & \frac{81x}{ \color{red}{81} }& = & \frac{740}{81} \\\Leftrightarrow & x = \frac{740}{81} & & \\ & V = \left\{ \frac{740}{81} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-30& = & 56x-126 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-56x} & = & \color{red}{56x} -126 \color{blue}{-56x} \color{blue}{+30} \\\Leftrightarrow & -35x& = & -96 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-96}{-35} \\\Leftrightarrow & x = \frac{96}{35} & & \\ & V = \left\{ \frac{96}{35} \right\} & \\\end{align}\)