Likninger med brøk — mathbusters

Oppgave 1

Løsningsforslag:
\begin{align*}
\frac{x-3}{4}&=\frac{49}{28}\\ \\
\frac{x-3}{4}&=\frac{7}{4}\\ \\
x-3&=7\\ \\
x&=10
\end{align*}
Løsningsforslag:
\begin{align*}
\frac{5+x}{10}&=\frac{2}{5}\\ \\
5+x&=4\\ \\
x&=-1
\end{align*}
Løsningsforslag:
\begin{align}
\frac{x}{30}&=\frac{7}{10} \hspace{1cm} |\cdot 10\\ \\
\frac{x}{3}&=7\\ \\
x&=21
\end{align}
Løsningsforslag:
\begin{align}
\frac{x-2}{16}&=\frac{x}{4}\\ \\
x-2&=4x\\ \\
3x&=-2\\ \\
x&=-\frac{2}{3}
\end{align}
Løsningsforslag:
\begin{align}
\frac{x}{2}&=\frac{6}{30}\\ \\
\frac{x}{2}&=\frac{1}{5}\\ \\
5x&=2\\ \\
x&=\frac{2}{5}
\end{align}
Løsningsforslag:
\begin{align}
\frac{x+1}{7}&=\frac{6}{14}\\ \\
2(x+1)&=6\\ \\
2x+2&=6\\ \\
2x&=4\\ \\
x&=2
\end{align}

Oppgave 2

Løsningsforslag:
\begin{align}
\frac{x}{15}&=\frac{5}{75}\\ \\
\frac{x}{15}&=\frac{1}{15}\\ \\
x&=1
\end{align}
Løsningsforslag:
\begin{align}
\frac{x}{20}&=\frac{2}{10}\\ \\
x&=4
\end{align}
Løsningsforslag:
\begin{align}
\frac{x}{6}&=\frac{x-3}{12}\\ \\
2x&=x-3\\ \\
x&=-3
\end{align}
Løsningsforslag:
\begin{align}
\frac{x}{5}&=\frac{12}{6}\\ \\
\frac{x}{5}&=2\\ \\
x&=10
\end{align}
Løsningsforslag:
\begin{align}
\frac{6}{x+5}&=\frac{18}{24}\\ \\
\frac{6}{x+5}&=\frac{3}{4}\\ \\
6\cdot 4&=3(x+5)\\ \\
24&=3x+15\\ \\
3x&=9\\ \\
x&=3
\end{align}
Løsningsforslag
\begin{align}
\frac{5}{15}&=\frac{x}{9}\\
\frac{1}{3}&=\frac{x}{9}\\
x&=3
\end{align}

Oppgave 3

Løsningsforslag:
\begin{align}
\frac{x+2}{10}&=\frac{5}{2}\\ \\
x+2&=5\cdot 5\\ \\
x&=25-2\\ \\
x&=23
\end{align}
Løsningsforslag:
\begin{align}
\frac{1}{2}&=\frac{5}{x}\\ \\
x&=10
\end{align}
Løsningsforslag:
\begin{align}
\frac{x-4}{3}&=\frac{2x}{7}\\ \\
7(x-4)&=2x\cdot 3\\ \\
7x-28&=6x\\ \\
7x-6x&=28\\ \\
x&=28
\end{align}
Løsningsforslag:
\begin{align}
\frac{x-1}{2}&=\frac{x}{3}\\ \\
3(x-1)&=2x\\ \\
3x-3&=2x\\ \\
x&=3
\end{align}
Løsningsforslag:
\begin{align}
\frac{2x}{3}&=40\\ \\
2x&=120\\ \\
x&=60
\end{align}
Løsningsforslag
\begin{align}
\frac{x}{3}&=\frac{12}{27}\\ \\
\frac{x}{3}&=\frac{4}{9}\\ \\
x&=\frac{4\cdot 3}{9}\\ \\
x&=\frac{4}{3}
\end{align}

Oppgave 4

Løsningsforslag:
\begin{align}
\frac{5}{2} x+3&=\frac{1}{2}x+15\\ \\
2\cdot\bigg(\frac{5}{2} x+3\bigg)&=2\cdot\bigg(\frac{1}{2}x+15\bigg)\\ \\
5x+6&=x+30\\ \\
4x&=24\\ \\
x&=6
\end{align}
Løsningsforslag:
\begin{align}
\frac{x}{2}-\frac{x}{3}&=\frac{5}{6}\\ \\
6\cdot\bigg(\frac{x}{2}-\frac{x}{3}\bigg)&=6\cdot\bigg(\frac{5}{6}\bigg)\\ \\
3x-2x&=5\\ \\
x&=5
\end{align}
Løsningsforslag:
\begin{align}
\frac{3x}{2}+5&=\frac{1}{3}-\frac{5x}{6}\\ \\
6\cdot\bigg(\frac{3x}{2}+5\bigg)&=6\cdot\bigg(\frac{1}{3}-\frac{5x}{6}\bigg)\\ \\
9x+30&=2-5x\\ \\
14x&=-28\\ \\
x&=-2
\end{align}
Løsningsforslag:
\begin{align}
\frac{2x}{3}+7&=\frac{x}{6}\\ \\
6\cdot\bigg(\frac{2x}{3}+7\bigg)&=6\cdot\bigg(\frac{x}{6}\bigg)\\ \\
4x+42&=x\\ \\
3x&=-42\\ \\
x&=-14
\end{align}
Løsningsforslag:
\begin{align}
\frac{2x-1}{3}&=\frac{7x}{5}-\frac{3x-1}{4}\\ \\
60\cdot\bigg(\frac{2x-1}{3}\bigg)&=60\cdot\bigg(\frac{7x}{5}-\frac{3x-1}{4}\bigg)\\ \\
20\cdot 2x-20&=12\cdot 7x-15(3x-1)\\ \\
40x-20&=84x-45x+15\\ \\
40x+45x-84x&=15+20\\ \\
x&=35
\end{align}
\begin{align}
\frac{11x-1}{27}-\frac{4-17x}{81}&=\frac{5x+2}{9}\\ \\
81\cdot\bigg(\frac{11x-1}{27}-\frac{4-17x}{81}\bigg)&=81\cdot\bigg(\frac{5x+2}{9}\bigg)\\ \\
3(11x-1)-(4-17x)&=9(5x+2)\\ \\
33x-3-4+17x&=45x+18\\ \\
5x&=25\\ \\
x&=5
\end{align}

Oppgave 5

Løsningsforslag:
\begin{align}
\frac{1}{6}\big(\frac{x}{2}-2\big)-3\big(\frac{x}{4}-1\big)&=\frac{2}{3}\\ \\
\frac{x}{12}-\frac{2}{6}-\frac{3x}{4}+3&=\frac{2}{3}\\ \\
\frac{x\cdot 12}{12}-\frac{2\cdot 12}{6}-\frac{3x\cdot 12}{4}+3\cdot 12&=\frac{2\cdot 12}{3}\\ \\
x-4-9x+36&=8\\ \\
x-9x&=8+4-36\\ \\
-8x&=-24\\ \\
x&=3
\end{align}
Løsningsforslag:
\begin{align}
6-\frac{1}{3}(2x+5)&=4\\ \\
6\cdot 3-\frac{1\cdot 3}{3}(2x+5)&=4\cdot 3\\ \\
18-(2x+5)&=12\\ \\
18-2x-5&=12\\ \\
-2x&=12-18+5\\ \\
-2x&=-1\\ \\
x&=\frac{1}{2}
\end{align}
Løsningsforslag:
\begin{align}
\frac{3}{5}\big(1-\frac{x}{2}\big)-\frac{x-2}{2}&=0\\ \\
\frac{3\cdot 10}{5}\big(1-\frac{x}{2}\big)-\frac{(x-2)\cdot 10}{2}&=0\cdot 10\\ \\
6\big(1-\frac{x}{2}\big)-5(x-2)&=0\\
6-3x-5x+10&=0\\ \\
3x+5x&=6+10\\ \\
8x&=16\\
x&=2
\end{align}
Løsningsforslag:
\begin{align}
\frac{3x-5}{2}&=x+4\\ \\
3x-5&=2(x+4)\\ \\
3x-5&=2x+8\\ \\
3x-2x&=8+5\\ \\
x&=13
\end{align}
Løsningsforslag:
\begin{align}
\frac{x-4}{4}+\frac{x-3}{12}&=\frac{x+2}{6}\\ \\
3(x-4)+(x-3)&=2(x+2)\\ \\
3x-12+x-3&=2x+4\\ \\
4x-2x&=4+12+3\\ \\
2x&=19\\ \\
x&=\frac{19}{2}
\end{align}

Oppgave 6

Løsningsforslag:
\begin{align}
2x-\frac{x+1}{2}+\frac{1}{3}(x+1)&=\frac{7}{3}\\ \\
6\cdot\bigg(2x-\frac{x+1}{2}+\frac{1}{3}(x+1)\bigg)&=6\cdot \bigg(\frac{7}{3}\bigg)\\ \\
12x-3(x+1)+2(x+1)&=15\\ \\
12x-3x-3+2x+2&=15\\ \\
11x&=15\\ \\
x&=\frac{15}{11}
\end{align}
Løsningsforslag:
\begin{align}
\frac{3}{2}-\frac{x-5}{2}&=\frac{5}{2}\\ \\
2\cdot\bigg(\frac{3}{2}-\frac{x-5}{2}\bigg)&=2\cdot\bigg(\frac{5}{2}\bigg)\\ \\
3-(x-5)&=5\\ \\
3-x+5&=5\\ \\
x&=3
\end{align}
Løsningsforslag:
\begin{align}
\frac{1}{3}x+2&=\frac{1}{2}x-\frac{1}{3}\\ \\
6\cdot\bigg(\frac{1}{3}x+2\bigg)&=6\cdot\bigg(\frac{1}{2}x-\frac{1}{3}\bigg)\\ \\
2x+12&=3x-2\\ \\
2x-3x&=-2-12\\ \\
x&=14
\end{align}
Løsningsforslag:
\begin{align}
\frac{x}{4}&=12,5\\ \\
4\cdot \frac{x}{4}&=4\cdot 12,5\\ \\
x&=50
\end{align}
Løsningsforslag:
\begin{align}
4\cdot\bigg(\frac{x}{4}+8\bigg)&=4\cdot 11 \\ \\
x+32&=44\\ \\
x&=44-32\\ \\
x&=12
\end{align}
\begin{align*}
7\cdot\bigg(\frac{4x}{7}-2\bigg)&=7\cdot 6\\ \\
4x-14&=42\\ \\
4x&=42+14\\ \\
4x&=56\\ \\
\frac{4x}{4}&=\frac{4\cdot 14}{4}\\ \\
x&=14
\end{align*}
\begin{align}
\frac{3x}{2}&=\frac{2}{10}+\frac{2x}{5}\\ \\
15x&=2+4x\\ \\
11x&=2\\ \\
x&=\frac{2}{11}
\end{align}