Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-128}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-9& = & -128x+240 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{+128x} & = & \color{red}{-128x} +240 \color{blue}{+128x} \color{blue}{+9} \\\Leftrightarrow & 140x& = & 249 \\\Leftrightarrow & \frac{140x}{ \color{red}{140} }& = & \frac{249}{140} \\\Leftrightarrow & x = \frac{249}{140} & & \\ & V = \left\{ \frac{249}{140} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{7}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 294x-840 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-294x} & = & \color{red}{294x} -840 \color{blue}{-294x} \color{blue}{+120} \\\Leftrightarrow & -259x& = & -720 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{-720}{-259} \\\Leftrightarrow & x = \frac{720}{259} & & \\ & V = \left\{ \frac{720}{259} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{14}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 15 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-15& = & 98x-336 \\\Leftrightarrow & 21x \color{red}{-15} \color{blue}{+15} \color{blue}{-98x} & = & \color{red}{98x} -336 \color{blue}{-98x} \color{blue}{+15} \\\Leftrightarrow & -77x& = & -321 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-321}{-77} \\\Leftrightarrow & x = \frac{321}{77} & & \\ & V = \left\{ \frac{321}{77} \right\} & \\\end{align}\)
4. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x-\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+25& = & -112x-160 \\\Leftrightarrow & 40x \color{red}{+25} \color{blue}{-25} \color{blue}{+112x} & = & \color{red}{-112x} -160 \color{blue}{+112x} \color{blue}{-25} \\\Leftrightarrow & 152x& = & -185 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{-185}{152} \\\Leftrightarrow & x = \frac{-185}{152} & & \\ & V = \left\{ \frac{-185}{152} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{2}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & 84x-840 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{-84x} & = & \color{red}{84x} -840 \color{blue}{-84x} \color{blue}{-90} \\\Leftrightarrow & -49x& = & -930 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-930}{-49} \\\Leftrightarrow & x = \frac{930}{49} & & \\ & V = \left\{ \frac{930}{49} \right\} & \\\end{align}\)
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. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{5}{2}x-1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x-\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 35x-14 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-35x} & = & \color{red}{35x} -14 \color{blue}{-35x} \color{blue}{+4} \\\Leftrightarrow & -33x& = & -10 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-10}{-33} \\\Leftrightarrow & x = \frac{10}{33} & & \\ & V = \left\{ \frac{10}{33} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & -312x+2340 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{+312x} & = & \color{red}{-312x} +2340 \color{blue}{+312x} \color{blue}{+60} \\\Leftrightarrow & 377x& = & 2400 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{2400}{377} \\\Leftrightarrow & x = \frac{2400}{377} & & \\ & V = \left\{ \frac{2400}{377} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{5}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 140x+588 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-140x} & = & \color{red}{140x} +588 \color{blue}{-140x} \color{blue}{-24} \\\Leftrightarrow & -119x& = & 564 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{564}{-119} \\\Leftrightarrow & x = \frac{-564}{119} & & \\ & V = \left\{ \frac{-564}{119} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{13}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-150& = & -546x-1170 \\\Leftrightarrow & 65x \color{red}{-150} \color{blue}{+150} \color{blue}{+546x} & = & \color{red}{-546x} -1170 \color{blue}{+546x} \color{blue}{+150} \\\Leftrightarrow & 611x& = & -1020 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{-1020}{611} \\\Leftrightarrow & x = \frac{-1020}{611} & & \\ & V = \left\{ \frac{-1020}{611} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{-7}{4}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & -147x-168 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{+147x} & = & \color{red}{-147x} -168 \color{blue}{+147x} \color{blue}{+24} \\\Leftrightarrow & 175x& = & -144 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{-144}{175} \\\Leftrightarrow & x = \frac{-144}{175} & & \\ & V = \left\{ \frac{-144}{175} \right\} & \\\end{align}\)
12. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{-44}{ \color{blue}{110} }x-\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-50& = & -44x-220 \\\Leftrightarrow & 55x \color{red}{-50} \color{blue}{+50} \color{blue}{+44x} & = & \color{red}{-44x} -220 \color{blue}{+44x} \color{blue}{+50} \\\Leftrightarrow & 99x& = & -170 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{-170}{99} \\\Leftrightarrow & x = \frac{-170}{99} & & \\ & V = \left\{ \frac{-170}{99} \right\} & \\\end{align}\)