Lineære likninger
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}