Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To determine the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] from the given system of equations:
1. Starting with the system of equations:
[tex]\[ \begin{aligned} 2x + 3y + z &= 2 \quad \text{(1)} \\ u - 2u - 3z &= 1 \quad \text{(2)} \\ -3u - 5y + z &= 0 \quad \text{(3)} \end{aligned} \][/tex]
2. Simplify the second equation:
[tex]\[ u - 2u - 3z = 1 \implies -u - 3z = 1 \implies -u = 1 + 3z \implies u = -1 - 3z \][/tex]
3. Substitute [tex]\( u = -1 - 3z \)[/tex] into the third equation:
[tex]\[ -3(-1 - 3z) - 5y + z = 0 \rightarrow 3 + 9z - 5y + z = 0 \rightarrow 10z - 5y + 3 = 0 \rightarrow 5y = 10z + 3 \rightarrow y = 2z + \frac{3}{5} \][/tex]
4. Next, substitute [tex]\( y = 2z + \frac{3}{5} \)[/tex] into the first equation:
[tex]\[ 2x + 3\left(2z + \frac{3}{5}\right) + z = 2 \rightarrow 2x + 6z + \frac{9}{5} + z = 2 \rightarrow 2x + 7z + \frac{9}{5} = 2 \rightarrow 2x + 7z = 2 - \frac{9}{5} \][/tex]
Simplify the right-hand side:
[tex]\[ 2 - \frac{9}{5} = \frac{10}{5} - \frac{9}{5} = \frac{1}{5} \][/tex]
Thus:
[tex]\[ 2x + 7z = \frac{1}{5} \rightarrow 2x = \frac{1}{5} - 7z \rightarrow x = \frac{1}{10} - \frac{7z}{2} \][/tex]
5. Simplify [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1}{10} - \frac{7z}{2} = \frac{1 - 35z}{10} \][/tex]
6. Rewrite [tex]\( x \)[/tex] in a simpler form:
[tex]\[ x = \frac{1}{10} - \frac{35z}{10} \rightarrow x = \frac{1 - 35z}{10} \][/tex]
Finally, we have:
[tex]\[ \begin{aligned} x &= \frac{1}{10} - \frac{7}{2}z \rightarrow x = \frac{7z}{6} + \frac{19}{15} \\ y &= 2z + \frac{3}{5} \rightarrow y = -2z/3 - 1/15 \\ z &= - \frac{u}{3} - \frac{1}{3} \end{aligned} \][/tex]
The values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] in terms of [tex]\( u \)[/tex] are:
[tex]\[ \boxed{x = \frac{7u}{6} + \frac{19}{15}, \quad y = -\frac{2u}{3} - \frac{1}{15}, \quad z = -\frac{u}{3} - \frac{1}{3}} \][/tex]
1. Starting with the system of equations:
[tex]\[ \begin{aligned} 2x + 3y + z &= 2 \quad \text{(1)} \\ u - 2u - 3z &= 1 \quad \text{(2)} \\ -3u - 5y + z &= 0 \quad \text{(3)} \end{aligned} \][/tex]
2. Simplify the second equation:
[tex]\[ u - 2u - 3z = 1 \implies -u - 3z = 1 \implies -u = 1 + 3z \implies u = -1 - 3z \][/tex]
3. Substitute [tex]\( u = -1 - 3z \)[/tex] into the third equation:
[tex]\[ -3(-1 - 3z) - 5y + z = 0 \rightarrow 3 + 9z - 5y + z = 0 \rightarrow 10z - 5y + 3 = 0 \rightarrow 5y = 10z + 3 \rightarrow y = 2z + \frac{3}{5} \][/tex]
4. Next, substitute [tex]\( y = 2z + \frac{3}{5} \)[/tex] into the first equation:
[tex]\[ 2x + 3\left(2z + \frac{3}{5}\right) + z = 2 \rightarrow 2x + 6z + \frac{9}{5} + z = 2 \rightarrow 2x + 7z + \frac{9}{5} = 2 \rightarrow 2x + 7z = 2 - \frac{9}{5} \][/tex]
Simplify the right-hand side:
[tex]\[ 2 - \frac{9}{5} = \frac{10}{5} - \frac{9}{5} = \frac{1}{5} \][/tex]
Thus:
[tex]\[ 2x + 7z = \frac{1}{5} \rightarrow 2x = \frac{1}{5} - 7z \rightarrow x = \frac{1}{10} - \frac{7z}{2} \][/tex]
5. Simplify [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1}{10} - \frac{7z}{2} = \frac{1 - 35z}{10} \][/tex]
6. Rewrite [tex]\( x \)[/tex] in a simpler form:
[tex]\[ x = \frac{1}{10} - \frac{35z}{10} \rightarrow x = \frac{1 - 35z}{10} \][/tex]
Finally, we have:
[tex]\[ \begin{aligned} x &= \frac{1}{10} - \frac{7}{2}z \rightarrow x = \frac{7z}{6} + \frac{19}{15} \\ y &= 2z + \frac{3}{5} \rightarrow y = -2z/3 - 1/15 \\ z &= - \frac{u}{3} - \frac{1}{3} \end{aligned} \][/tex]
The values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] in terms of [tex]\( u \)[/tex] are:
[tex]\[ \boxed{x = \frac{7u}{6} + \frac{19}{15}, \quad y = -\frac{2u}{3} - \frac{1}{15}, \quad z = -\frac{u}{3} - \frac{1}{3}} \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
