Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{4}-\frac{5}{12}=\frac{7}{3}x-8\)
2. \(\frac{x}{4}+\frac{2}{15}=\frac{-2}{3}x-3\)
3. \(\frac{x}{3}+\frac{4}{7}=\frac{-7}{4}x-1\)
4. \(\frac{x}{4}-\frac{3}{16}=\frac{-2}{3}x+4\)
5. \(\frac{x}{6}+\frac{3}{16}=\frac{6}{5}x+8\)
6. \(\frac{x}{7}-\frac{5}{8}=\frac{1}{6}x+8\)
7. \(\frac{x}{7}+\frac{4}{11}=\frac{3}{4}x+8\)
8. \(\frac{x}{6}+\frac{5}{6}=\frac{-7}{5}x+5\)
9. \(\frac{x}{2}+\frac{2}{13}=\frac{-2}{3}x-5\)
10. \(\frac{x}{7}-\frac{4}{7}=\frac{7}{2}x-4\)
11. \(\frac{x}{4}+\frac{2}{7}=\frac{-5}{3}x-2\)
12. \(\frac{x}{5}+\frac{4}{13}=\frac{1}{6}x-8\)

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

Verbetersleutel

1. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{12}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{28}{ \color{blue}{12} }x-\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-5& = & 28x-96 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-28x} & = & \color{red}{28x} -96 \color{blue}{-28x} \color{blue}{+5} \\\Leftrightarrow & -25x& = & -91 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-91}{-25} \\\Leftrightarrow & x = \frac{91}{25} & & \\ & V = \left\{ \frac{91}{25} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -40x-180 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+40x} & = & \color{red}{-40x} -180 \color{blue}{+40x} \color{blue}{-8} \\\Leftrightarrow & 55x& = & -188 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{-188}{55} \\\Leftrightarrow & x = \frac{-188}{55} & & \\ & V = \left\{ \frac{-188}{55} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{-7}{4}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & -147x-84 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{+147x} & = & \color{red}{-147x} -84 \color{blue}{+147x} \color{blue}{-48} \\\Leftrightarrow & 175x& = & -132 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{-132}{175} \\\Leftrightarrow & x = \frac{-132}{175} & & \\ & V = \left\{ \frac{-132}{175} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x+\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-9& = & -32x+192 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{+32x} & = & \color{red}{-32x} +192 \color{blue}{+32x} \color{blue}{+9} \\\Leftrightarrow & 44x& = & 201 \\\Leftrightarrow & \frac{44x}{ \color{red}{44} }& = & \frac{201}{44} \\\Leftrightarrow & x = \frac{201}{44} & & \\ & V = \left\{ \frac{201}{44} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x+\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & 288x+1920 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{-288x} & = & \color{red}{288x} +1920 \color{blue}{-288x} \color{blue}{-45} \\\Leftrightarrow & -248x& = & 1875 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{1875}{-248} \\\Leftrightarrow & x = \frac{-1875}{248} & & \\ & V = \left\{ \frac{-1875}{248} \right\} & \\\end{align}\)
6. \(\text{168 is het kleinste gemene veelvoud van 7, 8 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{8}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{168.} (\frac{24x}{ \color{blue}{168} }- \frac{ 105 }{ \color{blue}{168} })& = & (\frac{28}{ \color{blue}{168} }x+\frac{1344}{ \color{blue}{168} }) \color{blue}{.168} \\\Leftrightarrow & 24x-105& = & 28x+1344 \\\Leftrightarrow & 24x \color{red}{-105} \color{blue}{+105} \color{blue}{-28x} & = & \color{red}{28x} +1344 \color{blue}{-28x} \color{blue}{+105} \\\Leftrightarrow & -4x& = & 1449 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{1449}{-4} \\\Leftrightarrow & x = \frac{-1449}{4} & & \\ & V = \left\{ \frac{-1449}{4} \right\} & \\\end{align}\)
7. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{3}{4}x+8 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }+ \frac{ 112 }{ \color{blue}{308} })& = & (\frac{231}{ \color{blue}{308} }x+\frac{2464}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x+112& = & 231x+2464 \\\Leftrightarrow & 44x \color{red}{+112} \color{blue}{-112} \color{blue}{-231x} & = & \color{red}{231x} +2464 \color{blue}{-231x} \color{blue}{-112} \\\Leftrightarrow & -187x& = & 2352 \\\Leftrightarrow & \frac{-187x}{ \color{red}{-187} }& = & \frac{2352}{-187} \\\Leftrightarrow & x = \frac{-2352}{187} & & \\ & V = \left\{ \frac{-2352}{187} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -42x+150 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+42x} & = & \color{red}{-42x} +150 \color{blue}{+42x} \color{blue}{-25} \\\Leftrightarrow & 47x& = & 125 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{125}{47} \\\Leftrightarrow & x = \frac{125}{47} & & \\ & V = \left\{ \frac{125}{47} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & -52x-390 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{+52x} & = & \color{red}{-52x} -390 \color{blue}{+52x} \color{blue}{-12} \\\Leftrightarrow & 91x& = & -402 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-402}{91} \\\Leftrightarrow & x = \frac{-402}{91} & & \\ & V = \left\{ \frac{-402}{91} \right\} & \\\end{align}\)
10. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{7}{2}x-4 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x-\frac{56}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 49x-56 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-49x} & = & \color{red}{49x} -56 \color{blue}{-49x} \color{blue}{+8} \\\Leftrightarrow & -47x& = & -48 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{-48}{-47} \\\Leftrightarrow & x = \frac{48}{47} & & \\ & V = \left\{ \frac{48}{47} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & -140x-168 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+140x} & = & \color{red}{-140x} -168 \color{blue}{+140x} \color{blue}{-24} \\\Leftrightarrow & 161x& = & -192 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-192}{161} \\\Leftrightarrow & x = \frac{-192}{161} & & \\ & V = \left\{ \frac{-192}{161} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+120& = & 65x-3120 \\\Leftrightarrow & 78x \color{red}{+120} \color{blue}{-120} \color{blue}{-65x} & = & \color{red}{65x} -3120 \color{blue}{-65x} \color{blue}{-120} \\\Leftrightarrow & 13x& = & -3240 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-3240}{13} \\\Leftrightarrow & x = \frac{-3240}{13} & & \\ & V = \left\{ \frac{-3240}{13} \right\} & \\\end{align}\)