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

---

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

Verbetersleutel

1. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{3}{4}x-5 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 112 }{ \color{blue}{420} })& = & (\frac{315}{ \color{blue}{420} }x-\frac{2100}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+112& = & 315x-2100 \\\Leftrightarrow & 60x \color{red}{+112} \color{blue}{-112} \color{blue}{-315x} & = & \color{red}{315x} -2100 \color{blue}{-315x} \color{blue}{-112} \\\Leftrightarrow & -255x& = & -2212 \\\Leftrightarrow & \frac{-255x}{ \color{red}{-255} }& = & \frac{-2212}{-255} \\\Leftrightarrow & x = \frac{2212}{255} & & \\ & V = \left\{ \frac{2212}{255} \right\} & \\\end{align}\)
2. \(\text{168 is het kleinste gemene veelvoud van 7, 8 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{168.} (\frac{24x}{ \color{blue}{168} }+ \frac{ 63 }{ \color{blue}{168} })& = & (\frac{28}{ \color{blue}{168} }x+\frac{168}{ \color{blue}{168} }) \color{blue}{.168} \\\Leftrightarrow & 24x+63& = & 28x+168 \\\Leftrightarrow & 24x \color{red}{+63} \color{blue}{-63} \color{blue}{-28x} & = & \color{red}{28x} +168 \color{blue}{-28x} \color{blue}{-63} \\\Leftrightarrow & -4x& = & 105 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{105}{-4} \\\Leftrightarrow & x = \frac{-105}{4} & & \\ & V = \left\{ \frac{-105}{4} \right\} & \\\end{align}\)
3. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{-4}{ \color{blue}{10} }x+\frac{40}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x-3& = & -4x+40 \\\Leftrightarrow & 5x \color{red}{-3} \color{blue}{+3} \color{blue}{+4x} & = & \color{red}{-4x} +40 \color{blue}{+4x} \color{blue}{+3} \\\Leftrightarrow & 9x& = & 43 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{43}{9} \\\Leftrightarrow & x = \frac{43}{9} & & \\ & V = \left\{ \frac{43}{9} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & 112x+588 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-112x} & = & \color{red}{112x} +588 \color{blue}{-112x} \color{blue}{+18} \\\Leftrightarrow & -91x& = & 606 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{606}{-91} \\\Leftrightarrow & x = \frac{-606}{91} & & \\ & V = \left\{ \frac{-606}{91} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-336}{ \color{blue}{240} }x-\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -336x-1920 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+336x} & = & \color{red}{-336x} -1920 \color{blue}{+336x} \color{blue}{-75} \\\Leftrightarrow & 376x& = & -1995 \\\Leftrightarrow & \frac{376x}{ \color{red}{376} }& = & \frac{-1995}{376} \\\Leftrightarrow & x = \frac{-1995}{376} & & \\ & V = \left\{ \frac{-1995}{376} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -140x+84 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+140x} & = & \color{red}{-140x} +84 \color{blue}{+140x} \color{blue}{-48} \\\Leftrightarrow & 161x& = & 36 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{36}{161} \\\Leftrightarrow & x = \frac{36}{161} & & \\ & V = \left\{ \frac{36}{161} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{7}{5}x-4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{126}{ \color{blue}{90} }x-\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 126x-360 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-126x} & = & \color{red}{126x} -360 \color{blue}{-126x} \color{blue}{-40} \\\Leftrightarrow & -111x& = & -400 \\\Leftrightarrow & \frac{-111x}{ \color{red}{-111} }& = & \frac{-400}{-111} \\\Leftrightarrow & x = \frac{400}{111} & & \\ & V = \left\{ \frac{400}{111} \right\} & \\\end{align}\)
8. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{8}& = & \frac{5}{2}x+2 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }- \frac{ 25 }{ \color{blue}{40} })& = & (\frac{100}{ \color{blue}{40} }x+\frac{80}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x-25& = & 100x+80 \\\Leftrightarrow & 8x \color{red}{-25} \color{blue}{+25} \color{blue}{-100x} & = & \color{red}{100x} +80 \color{blue}{-100x} \color{blue}{+25} \\\Leftrightarrow & -92x& = & 105 \\\Leftrightarrow & \frac{-92x}{ \color{red}{-92} }& = & \frac{105}{-92} \\\Leftrightarrow & x = \frac{-105}{92} & & \\ & V = \left\{ \frac{-105}{92} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{14}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-45& = & 42x-840 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{-42x} & = & \color{red}{42x} -840 \color{blue}{-42x} \color{blue}{+45} \\\Leftrightarrow & -7x& = & -795 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-795}{-7} \\\Leftrightarrow & x = \frac{795}{7} & & \\ & V = \left\{ \frac{795}{7} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{3}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & 126x+840 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{-126x} & = & \color{red}{126x} +840 \color{blue}{-126x} \color{blue}{-150} \\\Leftrightarrow & -91x& = & 690 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{690}{-91} \\\Leftrightarrow & x = \frac{-690}{91} & & \\ & V = \left\{ \frac{-690}{91} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{7}{5}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{196}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-60& = & 196x-280 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-196x} & = & \color{red}{196x} -280 \color{blue}{-196x} \color{blue}{+60} \\\Leftrightarrow & -161x& = & -220 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{-220}{-161} \\\Leftrightarrow & x = \frac{220}{161} & & \\ & V = \left\{ \frac{220}{161} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x+\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & -52x+234 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{+52x} & = & \color{red}{-52x} +234 \color{blue}{+52x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & 216 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{216}{91} \\\Leftrightarrow & x = \frac{216}{91} & & \\ & V = \left\{ \frac{216}{91} \right\} & \\\end{align}\)