Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 30 }{ \color{blue}{195} })& = & (\frac{-520}{ \color{blue}{195} }x-\frac{1560}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+30& = & -520x-1560 \\\Leftrightarrow & 39x \color{red}{+30} \color{blue}{-30} \color{blue}{+520x} & = & \color{red}{-520x} -1560 \color{blue}{+520x} \color{blue}{-30} \\\Leftrightarrow & 559x& = & -1590 \\\Leftrightarrow & \frac{559x}{ \color{red}{559} }& = & \frac{-1590}{559} \\\Leftrightarrow & x = \frac{-1590}{559} & & \\ & V = \left\{ \frac{-1590}{559} \right\} & \\\end{align}\)
2. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 66x-1320 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-66x} & = & \color{red}{66x} -1320 \color{blue}{-66x} \color{blue}{+150} \\\Leftrightarrow & -11x& = & -1170 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-1170}{-11} \\\Leftrightarrow & x = \frac{1170}{11} & & \\ & V = \left\{ \frac{1170}{11} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & -28x+350 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+28x} & = & \color{red}{-28x} +350 \color{blue}{+28x} \color{blue}{+40} \\\Leftrightarrow & 63x& = & 390 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{390}{63} \\\Leftrightarrow & x = \frac{130}{21} & & \\ & V = \left\{ \frac{130}{21} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -312x+1170 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+312x} & = & \color{red}{-312x} +1170 \color{blue}{+312x} \color{blue}{-120} \\\Leftrightarrow & 377x& = & 1050 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{1050}{377} \\\Leftrightarrow & x = \frac{1050}{377} & & \\ & V = \left\{ \frac{1050}{377} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -56x+336 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+56x} & = & \color{red}{-56x} +336 \color{blue}{+56x} \color{blue}{-48} \\\Leftrightarrow & 77x& = & 288 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{288}{77} \\\Leftrightarrow & x = \frac{288}{77} & & \\ & V = \left\{ \frac{288}{77} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{7}{2}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 245x-350 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-245x} & = & \color{red}{245x} -350 \color{blue}{-245x} \color{blue}{+40} \\\Leftrightarrow & -231x& = & -310 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{-310}{-231} \\\Leftrightarrow & x = \frac{310}{231} & & \\ & V = \left\{ \frac{310}{231} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{14}& = & \frac{7}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 30 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-30& = & 196x-252 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-196x} & = & \color{red}{196x} -252 \color{blue}{-196x} \color{blue}{+30} \\\Leftrightarrow & -175x& = & -222 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-222}{-175} \\\Leftrightarrow & x = \frac{222}{175} & & \\ & V = \left\{ \frac{222}{175} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-20& = & 35x+140 \\\Leftrightarrow & 14x \color{red}{-20} \color{blue}{+20} \color{blue}{-35x} & = & \color{red}{35x} +140 \color{blue}{-35x} \color{blue}{+20} \\\Leftrightarrow & -21x& = & 160 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{160}{-21} \\\Leftrightarrow & x = \frac{-160}{21} & & \\ & V = \left\{ \frac{-160}{21} \right\} & \\\end{align}\)
9. \(\text{168 is het kleinste gemene veelvoud van 7, 8 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{8}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{168.} (\frac{24x}{ \color{blue}{168} }- \frac{ 63 }{ \color{blue}{168} })& = & (\frac{-112}{ \color{blue}{168} }x-\frac{168}{ \color{blue}{168} }) \color{blue}{.168} \\\Leftrightarrow & 24x-63& = & -112x-168 \\\Leftrightarrow & 24x \color{red}{-63} \color{blue}{+63} \color{blue}{+112x} & = & \color{red}{-112x} -168 \color{blue}{+112x} \color{blue}{+63} \\\Leftrightarrow & 136x& = & -105 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{-105}{136} \\\Leftrightarrow & x = \frac{-105}{136} & & \\ & V = \left\{ \frac{-105}{136} \right\} & \\\end{align}\)
10. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{-2}{5}x-4 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 80 }{ \color{blue}{220} })& = & (\frac{-88}{ \color{blue}{220} }x-\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+80& = & -88x-880 \\\Leftrightarrow & 55x \color{red}{+80} \color{blue}{-80} \color{blue}{+88x} & = & \color{red}{-88x} -880 \color{blue}{+88x} \color{blue}{-80} \\\Leftrightarrow & 143x& = & -960 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-960}{143} \\\Leftrightarrow & x = \frac{-960}{143} & & \\ & V = \left\{ \frac{-960}{143} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{2}{3}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 56x+252 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-56x} & = & \color{red}{56x} +252 \color{blue}{-56x} \color{blue}{+36} \\\Leftrightarrow & -35x& = & 288 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{288}{-35} \\\Leftrightarrow & x = \frac{-288}{35} & & \\ & V = \left\{ \frac{-288}{35} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{7}{5}x+2 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{112}{ \color{blue}{80} }x+\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & 112x+160 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{-112x} & = & \color{red}{112x} +160 \color{blue}{-112x} \color{blue}{-15} \\\Leftrightarrow & -92x& = & 145 \\\Leftrightarrow & \frac{-92x}{ \color{red}{-92} }& = & \frac{145}{-92} \\\Leftrightarrow & x = \frac{-145}{92} & & \\ & V = \left\{ \frac{-145}{92} \right\} & \\\end{align}\)