Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{8}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{-96}{ \color{blue}{120} }x+\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+45& = & -96x+600 \\\Leftrightarrow & 20x \color{red}{+45} \color{blue}{-45} \color{blue}{+96x} & = & \color{red}{-96x} +600 \color{blue}{+96x} \color{blue}{-45} \\\Leftrightarrow & 116x& = & 555 \\\Leftrightarrow & \frac{116x}{ \color{red}{116} }& = & \frac{555}{116} \\\Leftrightarrow & x = \frac{555}{116} & & \\ & V = \left\{ \frac{555}{116} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 42x+210 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-42x} & = & \color{red}{42x} +210 \color{blue}{-42x} \color{blue}{+90} \\\Leftrightarrow & -7x& = & 300 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{300}{-7} \\\Leftrightarrow & x = \frac{-300}{7} & & \\ & V = \left\{ \frac{-300}{7} \right\} & \\\end{align}\)
3. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{3}{4}x-7 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{18}{ \color{blue}{24} }x-\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & 18x-168 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{-18x} & = & \color{red}{18x} -168 \color{blue}{-18x} \color{blue}{-9} \\\Leftrightarrow & -10x& = & -177 \\\Leftrightarrow & \frac{-10x}{ \color{red}{-10} }& = & \frac{-177}{-10} \\\Leftrightarrow & x = \frac{177}{10} & & \\ & V = \left\{ \frac{177}{10} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 168x+420 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-168x} & = & \color{red}{168x} +420 \color{blue}{-168x} \color{blue}{+120} \\\Leftrightarrow & -133x& = & 540 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{540}{-133} \\\Leftrightarrow & x = \frac{-540}{133} & & \\ & V = \left\{ \frac{-540}{133} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x+\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & -264x+660 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{+264x} & = & \color{red}{-264x} +660 \color{blue}{+264x} \color{blue}{-120} \\\Leftrightarrow & 319x& = & 540 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{540}{319} \\\Leftrightarrow & x = \frac{540}{319} & & \\ & V = \left\{ \frac{540}{319} \right\} & \\\end{align}\)
6. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+20& = & 55x-550 \\\Leftrightarrow & 22x \color{red}{+20} \color{blue}{-20} \color{blue}{-55x} & = & \color{red}{55x} -550 \color{blue}{-55x} \color{blue}{-20} \\\Leftrightarrow & -33x& = & -570 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-570}{-33} \\\Leftrightarrow & x = \frac{190}{11} & & \\ & V = \left\{ \frac{190}{11} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 14 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{14}& = & \frac{2}{5}x+8 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 50 }{ \color{blue}{140} })& = & (\frac{56}{ \color{blue}{140} }x+\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+50& = & 56x+1120 \\\Leftrightarrow & 35x \color{red}{+50} \color{blue}{-50} \color{blue}{-56x} & = & \color{red}{56x} +1120 \color{blue}{-56x} \color{blue}{-50} \\\Leftrightarrow & -21x& = & 1070 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{1070}{-21} \\\Leftrightarrow & x = \frac{-1070}{21} & & \\ & V = \left\{ \frac{-1070}{21} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{3}{4}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & 63x+168 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{-63x} & = & \color{red}{63x} +168 \color{blue}{-63x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & 192 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{192}{-35} \\\Leftrightarrow & x = \frac{-192}{35} & & \\ & V = \left\{ \frac{-192}{35} \right\} & \\\end{align}\)
9. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x-\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+60& = & -84x-735 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+84x} & = & \color{red}{-84x} -735 \color{blue}{+84x} \color{blue}{-60} \\\Leftrightarrow & 119x& = & -795 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-795}{119} \\\Leftrightarrow & x = \frac{-795}{119} & & \\ & V = \left\{ \frac{-795}{119} \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{-110}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & -110x-396 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{+110x} & = & \color{red}{-110x} -396 \color{blue}{+110x} \color{blue}{+12} \\\Leftrightarrow & 143x& = & -384 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-384}{143} \\\Leftrightarrow & x = \frac{-384}{143} & & \\ & V = \left\{ \frac{-384}{143} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{9}& = & \frac{2}{5}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{36}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-20& = & 36x+180 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{-36x} & = & \color{red}{36x} +180 \color{blue}{-36x} \color{blue}{+20} \\\Leftrightarrow & -21x& = & 200 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{200}{-21} \\\Leftrightarrow & x = \frac{-200}{21} & & \\ & V = \left\{ \frac{-200}{21} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x+1050 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} +1050 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & 1110 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{1110}{203} \\\Leftrightarrow & x = \frac{1110}{203} & & \\ & V = \left\{ \frac{1110}{203} \right\} & \\\end{align}\)