Quick questions on Systems of linear equations and larger matrices in QCE Specialist Mathematics Unit 3

A system such as $$\begin{aligned} a_1 x + b_1 y + c_1 z &= d_1\\ a_2 x + b_2 y + c_2 z &= d_2\\ a_3 x + b_3 y + c_3 z &= d_3 \end{aligned}$$is written compactly as$A\mathbf{x} = \mathbf{b}$, where$A$is the$3\times 3$matrix of coefficients,$\mathbf{x} = (x, y, z)^{\mathsf T}$and$\mathbf{b} = (d_1, d_2, d_3)^{\mathsf T}$.

When the inverse is not used, Gaussian elimination is the systematic alternative. Use one equation to eliminate a variable from the others, then repeat, reducing the system to an upper-triangular form that can be back-substituted. If elimination produces a contradictory row such as $0 = 5$, the system is inconsistent and has no solution. If it produces an all-zero row such as $0 = 0$, one equation was dependent on the others, leaving a free variable and infinitely many solutions.

When there are infinitely many solutions, express the answer with a parameter. Solve for two variables in terms of the third, set the free variable equal to $t$, and write the solution as $\mathbf{x} = \mathbf{x}_0 + t\mathbf{d}$. Geometrically this is exactly the line of intersection of the three planes: $\mathbf{x}_0$ is one point on the line and $\mathbf{d}$ is its direction. Recognising the parametric solution as a line ties this dot point back to vector equations of lines, and QCAA rewards stating the geometric interpretation alongside the algebra.