Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{-7}{4}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & -147x+588 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{+147x} & = & \color{red}{-147x} +588 \color{blue}{+147x} \color{blue}{-48} \\\Leftrightarrow & 175x& = & 540 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{540}{175} \\\Leftrightarrow & x = \frac{108}{35} & & \\ & V = \left\{ \frac{108}{35} \right\} & \\\end{align}\)
2. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x-\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-25& = & -64x-160 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{+64x} & = & \color{red}{-64x} -160 \color{blue}{+64x} \color{blue}{+25} \\\Leftrightarrow & 84x& = & -135 \\\Leftrightarrow & \frac{84x}{ \color{red}{84} }& = & \frac{-135}{84} \\\Leftrightarrow & x = \frac{-45}{28} & & \\ & V = \left\{ \frac{-45}{28} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -224x-504 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+224x} & = & \color{red}{-224x} -504 \color{blue}{+224x} \color{blue}{+24} \\\Leftrightarrow & 245x& = & -480 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-480}{245} \\\Leftrightarrow & x = \frac{-96}{49} & & \\ & V = \left\{ \frac{-96}{49} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-48& = & 44x-792 \\\Leftrightarrow & 33x \color{red}{-48} \color{blue}{+48} \color{blue}{-44x} & = & \color{red}{44x} -792 \color{blue}{-44x} \color{blue}{+48} \\\Leftrightarrow & -11x& = & -744 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-744}{-11} \\\Leftrightarrow & x = \frac{744}{11} & & \\ & V = \left\{ \frac{744}{11} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & -264x+2640 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{+264x} & = & \color{red}{-264x} +2640 \color{blue}{+264x} \color{blue}{-90} \\\Leftrightarrow & 319x& = & 2550 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{2550}{319} \\\Leftrightarrow & x = \frac{2550}{319} & & \\ & V = \left\{ \frac{2550}{319} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & -264x+330 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{+264x} & = & \color{red}{-264x} +330 \color{blue}{+264x} \color{blue}{-120} \\\Leftrightarrow & 319x& = & 210 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{210}{319} \\\Leftrightarrow & x = \frac{210}{319} & & \\ & V = \left\{ \frac{210}{319} \right\} & \\\end{align}\)
7. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -312x+3120 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+312x} & = & \color{red}{-312x} +3120 \color{blue}{+312x} \color{blue}{-90} \\\Leftrightarrow & 377x& = & 3030 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{3030}{377} \\\Leftrightarrow & x = \frac{3030}{377} & & \\ & V = \left\{ \frac{3030}{377} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x+\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & 44x+264 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-44x} & = & \color{red}{44x} +264 \color{blue}{-44x} \color{blue}{-24} \\\Leftrightarrow & -11x& = & 240 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{240}{-11} \\\Leftrightarrow & x = \frac{-240}{11} & & \\ & V = \left\{ \frac{-240}{11} \right\} & \\\end{align}\)
9. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{11}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 30 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-30& = & 55x-770 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-55x} & = & \color{red}{55x} -770 \color{blue}{-55x} \color{blue}{+30} \\\Leftrightarrow & -33x& = & -740 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-740}{-33} \\\Leftrightarrow & x = \frac{740}{33} & & \\ & V = \left\{ \frac{740}{33} \right\} & \\\end{align}\)
10. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{5}{3}x+1 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 12 }{ \color{blue}{21} })& = & (\frac{35}{ \color{blue}{21} }x+\frac{21}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-12& = & 35x+21 \\\Leftrightarrow & 3x \color{red}{-12} \color{blue}{+12} \color{blue}{-35x} & = & \color{red}{35x} +21 \color{blue}{-35x} \color{blue}{+12} \\\Leftrightarrow & -32x& = & 33 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{33}{-32} \\\Leftrightarrow & x = \frac{-33}{32} & & \\ & V = \left\{ \frac{-33}{32} \right\} & \\\end{align}\)
11. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }- \frac{ 14 }{ \color{blue}{63} })& = & (\frac{-105}{ \color{blue}{63} }x+\frac{315}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x-14& = & -105x+315 \\\Leftrightarrow & 9x \color{red}{-14} \color{blue}{+14} \color{blue}{+105x} & = & \color{red}{-105x} +315 \color{blue}{+105x} \color{blue}{+14} \\\Leftrightarrow & 114x& = & 329 \\\Leftrightarrow & \frac{114x}{ \color{red}{114} }& = & \frac{329}{114} \\\Leftrightarrow & x = \frac{329}{114} & & \\ & V = \left\{ \frac{329}{114} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 18x-360 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-18x} & = & \color{red}{18x} -360 \color{blue}{-18x} \color{blue}{-40} \\\Leftrightarrow & -3x& = & -400 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-400}{-3} \\\Leftrightarrow & x = \frac{400}{3} & & \\ & V = \left\{ \frac{400}{3} \right\} & \\\end{align}\)