Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{28}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 28x+420 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-28x} & = & \color{red}{28x} +420 \color{blue}{-28x} \color{blue}{-80} \\\Leftrightarrow & 7x& = & 340 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{340}{7} \\\Leftrightarrow & x = \frac{340}{7} & & \\ & V = \left\{ \frac{340}{7} \right\} & \\\end{align}\)
2. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x+\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-60& = & 52x+936 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{-52x} & = & \color{red}{52x} +936 \color{blue}{-52x} \color{blue}{+60} \\\Leftrightarrow & -13x& = & 996 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{996}{-13} \\\Leftrightarrow & x = \frac{-996}{13} & & \\ & V = \left\{ \frac{-996}{13} \right\} & \\\end{align}\)
3. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{7}{2}x-2 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 70 }{ \color{blue}{126} })& = & (\frac{441}{ \color{blue}{126} }x-\frac{252}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-70& = & 441x-252 \\\Leftrightarrow & 18x \color{red}{-70} \color{blue}{+70} \color{blue}{-441x} & = & \color{red}{441x} -252 \color{blue}{-441x} \color{blue}{+70} \\\Leftrightarrow & -423x& = & -182 \\\Leftrightarrow & \frac{-423x}{ \color{red}{-423} }& = & \frac{-182}{-423} \\\Leftrightarrow & x = \frac{182}{423} & & \\ & V = \left\{ \frac{182}{423} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{4}x-4 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x-\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-80& = & 35x-560 \\\Leftrightarrow & 28x \color{red}{-80} \color{blue}{+80} \color{blue}{-35x} & = & \color{red}{35x} -560 \color{blue}{-35x} \color{blue}{+80} \\\Leftrightarrow & -7x& = & -480 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-480}{-7} \\\Leftrightarrow & x = \frac{480}{7} & & \\ & V = \left\{ \frac{480}{7} \right\} & \\\end{align}\)
5. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{5}{3}x-6 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{20}{ \color{blue}{12} }x-\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & 20x-72 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{-20x} & = & \color{red}{20x} -72 \color{blue}{-20x} \color{blue}{+10} \\\Leftrightarrow & -17x& = & -62 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{-62}{-17} \\\Leftrightarrow & x = \frac{62}{17} & & \\ & V = \left\{ \frac{62}{17} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 42x-1050 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-42x} & = & \color{red}{42x} -1050 \color{blue}{-42x} \color{blue}{+120} \\\Leftrightarrow & -7x& = & -930 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-930}{-7} \\\Leftrightarrow & x = \frac{930}{7} & & \\ & V = \left\{ \frac{930}{7} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & -126x+360 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+126x} & = & \color{red}{-126x} +360 \color{blue}{+126x} \color{blue}{-20} \\\Leftrightarrow & 141x& = & 340 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{340}{141} \\\Leftrightarrow & x = \frac{340}{141} & & \\ & V = \left\{ \frac{340}{141} \right\} & \\\end{align}\)
8. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x-\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & -32x-336 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{+32x} & = & \color{red}{-32x} -336 \color{blue}{+32x} \color{blue}{-15} \\\Leftrightarrow & 56x& = & -351 \\\Leftrightarrow & \frac{56x}{ \color{red}{56} }& = & \frac{-351}{56} \\\Leftrightarrow & x = \frac{-351}{56} & & \\ & V = \left\{ \frac{-351}{56} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 78x-3120 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-78x} & = & \color{red}{78x} -3120 \color{blue}{-78x} \color{blue}{-90} \\\Leftrightarrow & -13x& = & -3210 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-3210}{-13} \\\Leftrightarrow & x = \frac{3210}{13} & & \\ & V = \left\{ \frac{3210}{13} \right\} & \\\end{align}\)
10. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{-175}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+30& = & -175x+210 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{+175x} & = & \color{red}{-175x} +210 \color{blue}{+175x} \color{blue}{-30} \\\Leftrightarrow & 196x& = & 180 \\\Leftrightarrow & \frac{196x}{ \color{red}{196} }& = & \frac{180}{196} \\\Leftrightarrow & x = \frac{45}{49} & & \\ & V = \left\{ \frac{45}{49} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & 16x+48 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{-16x} & = & \color{red}{16x} +48 \color{blue}{-16x} \color{blue}{-9} \\\Leftrightarrow & -4x& = & 39 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{39}{-4} \\\Leftrightarrow & x = \frac{-39}{4} & & \\ & V = \left\{ \frac{-39}{4} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 7, 14 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x+25& = & 84x-140 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-84x} & = & \color{red}{84x} -140 \color{blue}{-84x} \color{blue}{-25} \\\Leftrightarrow & -74x& = & -165 \\\Leftrightarrow & \frac{-74x}{ \color{red}{-74} }& = & \frac{-165}{-74} \\\Leftrightarrow & x = \frac{165}{74} & & \\ & V = \left\{ \frac{165}{74} \right\} & \\\end{align}\)