Question

Suppose p = (1, 3, 4) and q = (5, 8, 12) are solutions to the linear system of equations: a11x1 + a12x2 + a13x3 = α1 a21x1 + a22x2 + a23x3 = α2 a31x1 + a32x2 + a33x3 = α3 a41x1 + a42x2 + a43x3 = α4 Check that all points on the line joining p and q are also solutions to the above system of equations. General Statement: If a system of linear equations in n variables has two solutions, then all points on the line joining the two solutions are also solutions to the system. Therefore, if a system of linear equations has at least two

Answer 1

Answer:

Let $A=\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\a_{41}&a_{42}&a_{43}\end{array}\right]$ the matrix associated to the system and

$b=\left[\begin{array}{ccc}\alpha_{1}\\\alpha_{2}\\\alpha_{3}\\\alpha_4\end{array}\right]$ the vector of constant values of the system.

The equation of the line joining p and q is

$p+t(q-p), t\in\mathbb{R}$. Since p and q are solutions of the linear system then, $Ap=b$ and $Aq=b$

Let $w=q-p$. Observe that $Aw=A(q-p)=Aq-Ap=b-b=0$. Then w is a solution of the homogeneous system Ax=0.

Now, let s=p+r(q-p) for some $r\in\mathbb{R}$  be a point in the line joining p and q. Observe that

$As=A(p+r(q-p))=Ap+rA(q-p)=Ap+rAw=b+r*0=b$

Then As=b. This means that s is a solution of the linear system.