Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{168 is het kleinste gemene veelvoud van 7, 8 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{8}& = & \frac{7}{6}x+7 \\\Leftrightarrow & \color{blue}{168.} (\frac{24x}{ \color{blue}{168} }+ \frac{ 105 }{ \color{blue}{168} })& = & (\frac{196}{ \color{blue}{168} }x+\frac{1176}{ \color{blue}{168} }) \color{blue}{.168} \\\Leftrightarrow & 24x+105& = & 196x+1176 \\\Leftrightarrow & 24x \color{red}{+105} \color{blue}{-105} \color{blue}{-196x} & = & \color{red}{196x} +1176 \color{blue}{-196x} \color{blue}{-105} \\\Leftrightarrow & -172x& = & 1071 \\\Leftrightarrow & \frac{-172x}{ \color{red}{-172} }& = & \frac{1071}{-172} \\\Leftrightarrow & x = \frac{-1071}{172} & & \\ & V = \left\{ \frac{-1071}{172} \right\} & \\\end{align}\)
2. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{3}{5}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{84}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+60& = & 84x-280 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-84x} & = & \color{red}{84x} -280 \color{blue}{-84x} \color{blue}{-60} \\\Leftrightarrow & -49x& = & -340 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-340}{-49} \\\Leftrightarrow & x = \frac{340}{49} & & \\ & V = \left\{ \frac{340}{49} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{-7}{4}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+24& = & -147x-336 \\\Leftrightarrow & 28x \color{red}{+24} \color{blue}{-24} \color{blue}{+147x} & = & \color{red}{-147x} -336 \color{blue}{+147x} \color{blue}{-24} \\\Leftrightarrow & 175x& = & -360 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{-360}{175} \\\Leftrightarrow & x = \frac{-72}{35} & & \\ & V = \left\{ \frac{-72}{35} \right\} & \\\end{align}\)
4. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 80 }{ \color{blue}{220} })& = & (\frac{-176}{ \color{blue}{220} }x+\frac{1320}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-80& = & -176x+1320 \\\Leftrightarrow & 55x \color{red}{-80} \color{blue}{+80} \color{blue}{+176x} & = & \color{red}{-176x} +1320 \color{blue}{+176x} \color{blue}{+80} \\\Leftrightarrow & 231x& = & 1400 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{1400}{231} \\\Leftrightarrow & x = \frac{200}{33} & & \\ & V = \left\{ \frac{200}{33} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }+ \frac{ 21 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x+21& = & -56x-560 \\\Leftrightarrow & 10x \color{red}{+21} \color{blue}{-21} \color{blue}{+56x} & = & \color{red}{-56x} -560 \color{blue}{+56x} \color{blue}{-21} \\\Leftrightarrow & 66x& = & -581 \\\Leftrightarrow & \frac{66x}{ \color{red}{66} }& = & \frac{-581}{66} \\\Leftrightarrow & x = \frac{-581}{66} & & \\ & V = \left\{ \frac{-581}{66} \right\} & \\\end{align}\)
6. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 105 }{ \color{blue}{231} })& = & (\frac{-616}{ \color{blue}{231} }x-\frac{1386}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-105& = & -616x-1386 \\\Leftrightarrow & 33x \color{red}{-105} \color{blue}{+105} \color{blue}{+616x} & = & \color{red}{-616x} -1386 \color{blue}{+616x} \color{blue}{+105} \\\Leftrightarrow & 649x& = & -1281 \\\Leftrightarrow & \frac{649x}{ \color{red}{649} }& = & \frac{-1281}{649} \\\Leftrightarrow & x = \frac{-1281}{649} & & \\ & V = \left\{ \frac{-1281}{649} \right\} & \\\end{align}\)
7. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x-\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & -156x-780 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{+156x} & = & \color{red}{-156x} -780 \color{blue}{+156x} \color{blue}{+120} \\\Leftrightarrow & 221x& = & -660 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-660}{221} \\\Leftrightarrow & x = \frac{-660}{221} & & \\ & V = \left\{ \frac{-660}{221} \right\} & \\\end{align}\)
8. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{8}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{32}{ \color{blue}{24} }x+\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-9& = & 32x+192 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{-32x} & = & \color{red}{32x} +192 \color{blue}{-32x} \color{blue}{+9} \\\Leftrightarrow & -20x& = & 201 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{201}{-20} \\\Leftrightarrow & x = \frac{-201}{20} & & \\ & V = \left\{ \frac{-201}{20} \right\} & \\\end{align}\)
9. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{8}& = & \frac{5}{6}x-6 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{100}{ \color{blue}{120} }x-\frac{720}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x-75& = & 100x-720 \\\Leftrightarrow & 24x \color{red}{-75} \color{blue}{+75} \color{blue}{-100x} & = & \color{red}{100x} -720 \color{blue}{-100x} \color{blue}{+75} \\\Leftrightarrow & -76x& = & -645 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{-645}{-76} \\\Leftrightarrow & x = \frac{645}{76} & & \\ & V = \left\{ \frac{645}{76} \right\} & \\\end{align}\)
10. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x-\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-25& = & -64x-480 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{+64x} & = & \color{red}{-64x} -480 \color{blue}{+64x} \color{blue}{+25} \\\Leftrightarrow & 84x& = & -455 \\\Leftrightarrow & \frac{84x}{ \color{red}{84} }& = & \frac{-455}{84} \\\Leftrightarrow & x = \frac{-65}{12} & & \\ & V = \left\{ \frac{-65}{12} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & 35x+210 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{-35x} & = & \color{red}{35x} +210 \color{blue}{-35x} \color{blue}{-120} \\\Leftrightarrow & 7x& = & 90 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{90}{7} \\\Leftrightarrow & x = \frac{90}{7} & & \\ & V = \left\{ \frac{90}{7} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-12& = & 39x+624 \\\Leftrightarrow & 26x \color{red}{-12} \color{blue}{+12} \color{blue}{-39x} & = & \color{red}{39x} +624 \color{blue}{-39x} \color{blue}{+12} \\\Leftrightarrow & -13x& = & 636 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{636}{-13} \\\Leftrightarrow & x = \frac{-636}{13} & & \\ & V = \left\{ \frac{-636}{13} \right\} & \\\end{align}\)