Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Solve the system of equations                         x + y + z = 6 <

Solve the system of equations

                        x + y + z = 6

                        x + 2y + 3z = 14

                        x + 4y + 7z = 30

Answers (1)

best_answer

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\textbf{Solve the system of equations:}
\[
\begin{aligned}
(1)\quad & x + y + z = 6 \\
(2)\quad & x + 2y + 3z = 14 \\
(3)\quad & x + 4y + 7z = 30
\end{aligned}
\]

\textbf{Step 1: Subtract equation (1) from equations (2) and (3)}

Subtracting (1) from (2):
\[
(x + 2y + 3z) - (x + y + z) = 14 - 6
\Rightarrow y + 2z = 8 \tag{4}
\]

Subtracting (1) from (3):
\[
(x + 4y + 7z) - (x + y + z) = 30 - 6
\Rightarrow 3y + 6z = 24 \tag{5}
\]

\textbf{Step 2: Solve equations (4) and (5)}

Equation (5) simplifies:
\[
3y + 6z = 24 \Rightarrow y + 2z = 8 \tag{5'}
\]

Now, equations (4) and (5') are the same:
\[
y + 2z = 8
\]

So, the system has infinitely many solutions. Let \( z = t \), a parameter.

Then,
\[
y = 8 - 2t
\]

\textbf{Step 3: Substitute into equation (1)}

Using equation (1): \( x + y + z = 6 \)
\[
x + (8 - 2t) + t = 6
\Rightarrow x + 8 - t = 6
\Rightarrow x = -2 + t
\]

\textbf{Final Answer:} (in terms of parameter \( t \))
\[
\boxed{
\begin{aligned}
x &= -2 + t \\
y &= 8 - 2t \\
z &= t
\end{aligned}
}
\quad \text{where } t \in \mathbb{R}
\]

\end{document}
 

Posted by

Saumya Singh

View full answer

Similar Questions

Trending Articles/News

anam-khan

CBSE Class 10 board exams twice a year ‘relief’ or ‘burden’? Parents, schools share mixed views
anam-khan

CBSE Class 10 exams twice a year from 2026; 64.4% students in favor, second attempt optional

Latest Question