If you have worked steadily through the Programme, you...

If you have worked steadily through the Programme, you should have no difficulty with this exercise. Take your time and work carefully. There is no extra credit for speed. Off you go then. They are all quite straightforward. By determinants, find the value of $x$, given: $$ \left\{\begin{array}{c} 2 x+3 y-z+3=0 \\ x-4 y+2 z-14=0 \\ 4 x+2 y-3 z+6=0 \end{array}\right. $$

If you have worked steadily through the Programme, you should have no difficulty with this exercise. Take your time and work carefully. There is no extra credit for speed. Off you go then. They are all quite straightforward.
By determinants, find the value of $x$, given:
$$
\left\{\begin{array}{c}
2 x+3 y-z+3=0 \\
x-4 y+2 z-14=0 \\
4 x+2 y-3 z+6=0
\end{array}\right.
$$

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Key Concepts

System of Linear Equations

A system of linear equations involves finding the values of unknown variables that simultaneously satisfy all the linear relationships given by the equations. In a standard setup, each equation represents a plane (in three variables) or line (in two variables) in a multi-dimensional space, and the solution is the point or set of points where these geometrical figures intersect.

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties of the matrix, such as whether it is invertible or singular. In the context of solving systems of equations, the determinant of the coefficient matrix indicates if a unique solution exists; a nonzero determinant shows that the system has a unique solution.

Cramer’s Rule

Cramer’s Rule is a method for solving systems of linear equations with the same number of equations as unknowns, provided the determinant of the coefficient matrix is nonzero. The rule expresses each variable as the ratio of two determinants: one determinant is of the original coefficient matrix, and the other is formed by replacing the appropriate column with the constant terms from the equations. This rule is particularly useful when dealing with small systems or when an explicit formula is required.

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