Vgln. eerste graad (reeks 4)

Hoofdmenu Eentje per keer  🖨

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

1. \(\frac{x}{4}+\frac{2}{15}=\frac{-2}{3}x-8\)
2. \(\frac{x}{3}+\frac{5}{14}=\frac{-2}{5}x-1\)
3. \(\frac{x}{5}-\frac{4}{13}=\frac{7}{6}x+7\)
4. \(\frac{x}{7}+\frac{5}{8}=\frac{7}{2}x-8\)
5. \(\frac{x}{4}+\frac{2}{9}=\frac{-2}{5}x+4\)
6. \(\frac{x}{2}+\frac{4}{11}=\frac{1}{5}x+8\)
7. \(\frac{x}{7}-\frac{2}{7}=\frac{3}{2}x+2\)
8. \(\frac{x}{7}-\frac{5}{16}=\frac{-3}{4}x+5\)
9. \(\frac{x}{5}-\frac{2}{11}=\frac{1}{2}x-3\)
10. \(\frac{x}{7}+\frac{2}{15}=\frac{1}{6}x+7\)
11. \(\frac{x}{3}+\frac{2}{11}=\frac{1}{2}x+8\)
12. \(\frac{x}{4}-\frac{3}{14}=\frac{-2}{3}x+5\)

---

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -40x-480 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+40x} & = & \color{red}{-40x} -480 \color{blue}{+40x} \color{blue}{-8} \\\Leftrightarrow & 55x& = & -488 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{-488}{55} \\\Leftrightarrow & x = \frac{-488}{55} & & \\ & V = \left\{ \frac{-488}{55} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 3, 14 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{14}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{70x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 70x+75& = & -84x-210 \\\Leftrightarrow & 70x \color{red}{+75} \color{blue}{-75} \color{blue}{+84x} & = & \color{red}{-84x} -210 \color{blue}{+84x} \color{blue}{-75} \\\Leftrightarrow & 154x& = & -285 \\\Leftrightarrow & \frac{154x}{ \color{red}{154} }& = & \frac{-285}{154} \\\Leftrightarrow & x = \frac{-285}{154} & & \\ & V = \left\{ \frac{-285}{154} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{7}{6}x+7 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{455}{ \color{blue}{390} }x+\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-120& = & 455x+2730 \\\Leftrightarrow & 78x \color{red}{-120} \color{blue}{+120} \color{blue}{-455x} & = & \color{red}{455x} +2730 \color{blue}{-455x} \color{blue}{+120} \\\Leftrightarrow & -377x& = & 2850 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{2850}{-377} \\\Leftrightarrow & x = \frac{-2850}{377} & & \\ & V = \left\{ \frac{-2850}{377} \right\} & \\\end{align}\)
4. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{8}& = & \frac{7}{2}x-8 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 35 }{ \color{blue}{56} })& = & (\frac{196}{ \color{blue}{56} }x-\frac{448}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+35& = & 196x-448 \\\Leftrightarrow & 8x \color{red}{+35} \color{blue}{-35} \color{blue}{-196x} & = & \color{red}{196x} -448 \color{blue}{-196x} \color{blue}{-35} \\\Leftrightarrow & -188x& = & -483 \\\Leftrightarrow & \frac{-188x}{ \color{red}{-188} }& = & \frac{-483}{-188} \\\Leftrightarrow & x = \frac{483}{188} & & \\ & V = \left\{ \frac{483}{188} \right\} & \\\end{align}\)
5. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-72}{ \color{blue}{180} }x+\frac{720}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+40& = & -72x+720 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{+72x} & = & \color{red}{-72x} +720 \color{blue}{+72x} \color{blue}{-40} \\\Leftrightarrow & 117x& = & 680 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{680}{117} \\\Leftrightarrow & x = \frac{680}{117} & & \\ & V = \left\{ \frac{680}{117} \right\} & \\\end{align}\)
6. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{22}{ \color{blue}{110} }x+\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+40& = & 22x+880 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{-22x} & = & \color{red}{22x} +880 \color{blue}{-22x} \color{blue}{-40} \\\Leftrightarrow & 33x& = & 840 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{840}{33} \\\Leftrightarrow & x = \frac{280}{11} & & \\ & V = \left\{ \frac{280}{11} \right\} & \\\end{align}\)
7. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{3}{2}x+2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x+\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 21x+28 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-21x} & = & \color{red}{21x} +28 \color{blue}{-21x} \color{blue}{+4} \\\Leftrightarrow & -19x& = & 32 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{32}{-19} \\\Leftrightarrow & x = \frac{-32}{19} & & \\ & V = \left\{ \frac{-32}{19} \right\} & \\\end{align}\)
8. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{-3}{4}x+5 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{-84}{ \color{blue}{112} }x+\frac{560}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & -84x+560 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{+84x} & = & \color{red}{-84x} +560 \color{blue}{+84x} \color{blue}{+35} \\\Leftrightarrow & 100x& = & 595 \\\Leftrightarrow & \frac{100x}{ \color{red}{100} }& = & \frac{595}{100} \\\Leftrightarrow & x = \frac{119}{20} & & \\ & V = \left\{ \frac{119}{20} \right\} & \\\end{align}\)
9. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 20 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{330}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-20& = & 55x-330 \\\Leftrightarrow & 22x \color{red}{-20} \color{blue}{+20} \color{blue}{-55x} & = & \color{red}{55x} -330 \color{blue}{-55x} \color{blue}{+20} \\\Leftrightarrow & -33x& = & -310 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-310}{-33} \\\Leftrightarrow & x = \frac{310}{33} & & \\ & V = \left\{ \frac{310}{33} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{1}{6}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 28 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+28& = & 35x+1470 \\\Leftrightarrow & 30x \color{red}{+28} \color{blue}{-28} \color{blue}{-35x} & = & \color{red}{35x} +1470 \color{blue}{-35x} \color{blue}{-28} \\\Leftrightarrow & -5x& = & 1442 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{1442}{-5} \\\Leftrightarrow & x = \frac{-1442}{5} & & \\ & V = \left\{ \frac{-1442}{5} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+12& = & 33x+528 \\\Leftrightarrow & 22x \color{red}{+12} \color{blue}{-12} \color{blue}{-33x} & = & \color{red}{33x} +528 \color{blue}{-33x} \color{blue}{-12} \\\Leftrightarrow & -11x& = & 516 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{516}{-11} \\\Leftrightarrow & x = \frac{-516}{11} & & \\ & V = \left\{ \frac{-516}{11} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & -56x+420 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+56x} & = & \color{red}{-56x} +420 \color{blue}{+56x} \color{blue}{+18} \\\Leftrightarrow & 77x& = & 438 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{438}{77} \\\Leftrightarrow & x = \frac{438}{77} & & \\ & V = \left\{ \frac{438}{77} \right\} & \\\end{align}\)