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

---

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

Verbetersleutel

1. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{4}{3}x+6 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 84 }{ \color{blue}{231} })& = & (\frac{308}{ \color{blue}{231} }x+\frac{1386}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-84& = & 308x+1386 \\\Leftrightarrow & 33x \color{red}{-84} \color{blue}{+84} \color{blue}{-308x} & = & \color{red}{308x} +1386 \color{blue}{-308x} \color{blue}{+84} \\\Leftrightarrow & -275x& = & 1470 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{1470}{-275} \\\Leftrightarrow & x = \frac{-294}{55} & & \\ & V = \left\{ \frac{-294}{55} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{13}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 30 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+30& = & 39x+546 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-39x} & = & \color{red}{39x} +546 \color{blue}{-39x} \color{blue}{-30} \\\Leftrightarrow & -13x& = & 516 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{516}{-13} \\\Leftrightarrow & x = \frac{-516}{13} & & \\ & V = \left\{ \frac{-516}{13} \right\} & \\\end{align}\)
3. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{-7}{4}x-2 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }+ \frac{ 84 }{ \color{blue}{364} })& = & (\frac{-637}{ \color{blue}{364} }x-\frac{728}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x+84& = & -637x-728 \\\Leftrightarrow & 52x \color{red}{+84} \color{blue}{-84} \color{blue}{+637x} & = & \color{red}{-637x} -728 \color{blue}{+637x} \color{blue}{-84} \\\Leftrightarrow & 689x& = & -812 \\\Leftrightarrow & \frac{689x}{ \color{red}{689} }& = & \frac{-812}{689} \\\Leftrightarrow & x = \frac{-812}{689} & & \\ & V = \left\{ \frac{-812}{689} \right\} & \\\end{align}\)
4. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }- \frac{ 56 }{ \color{blue}{308} })& = & (\frac{-231}{ \color{blue}{308} }x+\frac{2464}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x-56& = & -231x+2464 \\\Leftrightarrow & 44x \color{red}{-56} \color{blue}{+56} \color{blue}{+231x} & = & \color{red}{-231x} +2464 \color{blue}{+231x} \color{blue}{+56} \\\Leftrightarrow & 275x& = & 2520 \\\Leftrightarrow & \frac{275x}{ \color{red}{275} }& = & \frac{2520}{275} \\\Leftrightarrow & x = \frac{504}{55} & & \\ & V = \left\{ \frac{504}{55} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & 80x+480 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{-80x} & = & \color{red}{80x} +480 \color{blue}{-80x} \color{blue}{+18} \\\Leftrightarrow & -65x& = & 498 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{498}{-65} \\\Leftrightarrow & x = \frac{-498}{65} & & \\ & V = \left\{ \frac{-498}{65} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{1}{6}x-2 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x-\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & 40x-480 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{-40x} & = & \color{red}{40x} -480 \color{blue}{-40x} \color{blue}{+45} \\\Leftrightarrow & 8x& = & -435 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{-435}{8} \\\Leftrightarrow & x = \frac{-435}{8} & & \\ & V = \left\{ \frac{-435}{8} \right\} & \\\end{align}\)
7. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -312x-390 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+312x} & = & \color{red}{-312x} -390 \color{blue}{+312x} \color{blue}{-90} \\\Leftrightarrow & 377x& = & -480 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-480}{377} \\\Leftrightarrow & x = \frac{-480}{377} & & \\ & V = \left\{ \frac{-480}{377} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & 42x+1050 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{-42x} & = & \color{red}{42x} +1050 \color{blue}{-42x} \color{blue}{-150} \\\Leftrightarrow & -7x& = & 900 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{900}{-7} \\\Leftrightarrow & x = \frac{-900}{7} & & \\ & V = \left\{ \frac{-900}{7} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-30& = & -56x-210 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{+56x} & = & \color{red}{-56x} -210 \color{blue}{+56x} \color{blue}{+30} \\\Leftrightarrow & 91x& = & -180 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-180}{91} \\\Leftrightarrow & x = \frac{-180}{91} & & \\ & V = \left\{ \frac{-180}{91} \right\} & \\\end{align}\)
10. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{-44}{ \color{blue}{110} }x+\frac{330}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+20& = & -44x+330 \\\Leftrightarrow & 55x \color{red}{+20} \color{blue}{-20} \color{blue}{+44x} & = & \color{red}{-44x} +330 \color{blue}{+44x} \color{blue}{-20} \\\Leftrightarrow & 99x& = & 310 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{310}{99} \\\Leftrightarrow & x = \frac{310}{99} & & \\ & V = \left\{ \frac{310}{99} \right\} & \\\end{align}\)
11. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{12}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{28}{ \color{blue}{12} }x+\frac{24}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+5& = & 28x+24 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-28x} & = & \color{red}{28x} +24 \color{blue}{-28x} \color{blue}{-5} \\\Leftrightarrow & -25x& = & 19 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{19}{-25} \\\Leftrightarrow & x = \frac{-19}{25} & & \\ & V = \left\{ \frac{-19}{25} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{7}{2}x-6 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x-\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-30& = & 245x-420 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-245x} & = & \color{red}{245x} -420 \color{blue}{-245x} \color{blue}{+30} \\\Leftrightarrow & -231x& = & -390 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{-390}{-231} \\\Leftrightarrow & x = \frac{130}{77} & & \\ & V = \left\{ \frac{130}{77} \right\} & \\\end{align}\)