UPSKILL MATH PLUS

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Learn moreLet us learn how to find the solution to the system of linear equations in three variables.

Example:

Solve the system of linear equations in three variables \(6x + 4y - 2z = 12\), \(-2x + 2y + z = 3\), \(2x + 2y + 2z = 8\).

**Solution**:

Let us name the equations.

\(6x + 4y - 2z = 12\) ---- (\(1\))

\(-2x + 2y + z = 3\) ---- (\(2\))

\(2x + 2y + 2z = 8\) ---- (\(3\))

**Step 1**: Solving equations (\(2\)) and (\(3\)).

\(-2x + 2y + z = 3\)

\(2x + 2y + 2z = 8\)

__________________________

\(4y + 3z = 11\) ---- (\(4\))

**Step 2**: Similarly, let us eliminate the variable \(x\) from equations \(1\) and \(2\).

\(6x + 4y -2z = 12\)

\(-6x + 6y + 3z = 9\)

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\(10y + z = 21\) ---- (\(5\))

**Step 3**: Solve equations (\(4\)) and (\(5\)).

\(4y + 3z = 11\)

\(30y + 3z = 63\)

(\(-\)) (\(-\)) (\(-\))

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\(- 26y = - 52\)

\(y = 2\)

Substitute the value of \(y\) in equation (\(5\)), we get:

\(20 + z = 21\)

\(z = 1\)

**Step 4**: Substitute the value of \(y\) and \(z\) in equation (\(1\)), we get:

\(6x + 4(2) - 2(1) = 12\)

\(6x + 8 - 2 = 12\)

\(6x + 6 = 12\)

\(6x = 6\)

\(x = 1\)

Therefore, the solution is \(x = 1\), \(y = 2\) and \(z = 1\).