Problem 47 - Algebra and Trigonometry

Question

Can a Linear System Have Exactly Two Solutions? (a) Suppose that \(\left(x_{0}, y_{0}, z_{0}\right)\) and \(\left(x_{1}, y_{1}, z_{1}\right)\) are solutions of the system $$ \begin{array}{l}{\qquad\left\\{\begin{array}{l}{a_{1} x+b_{1} y+c_{1} z=d_{1}} \\\ {a_{2} x+b_{2} y+c_{2} z=d_{2}} \\ {a_{3} x+b_{3} y+c_{3} z=d_{3}}\end{array}\right.} \\ {\text { Show that }\left(\frac{x_{0}+x_{1}}{2}, \frac{y_{0}+y_{1}}{2}, \frac{z_{0}+z_{1}}{2}\right) \text { is also a solution. }}\end{array} $$ (b) Use the result of part (a) to prove that if the system has two different solutions, then it has infinitely many solutions.

Step-by-Step Solution

Answer

No, a linear system cannot have exactly two solutions; it has either none, one, or infinitely many.

Step 1: Assume Given Solutions Are Valid

We start by assuming that \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\) are solutions to the linear system:\[\begin{array}{l} a_1 x + b_1 y + c_1 z = d_1 \ a_2 x + b_2 y + c_2 z = d_2 \ a_3 x + b_3 y + c_3 z = d_3\end{array}\]This means they satisfy all three equations individually.

Step 2: Calculate Midpoint of the Solutions

Calculate the midpoint of the two solutions: \[\left(\frac{x_0 + x_1}{2}, \frac{y_0 + y_1}{2}, \frac{z_0 + z_1}{2}\right)\]This midpoint will be our candidate solution that we need to verify in the equations.

Step 3: Substitute Midpoint into the System

Substitute the midpoint into each equation of the system to check if it satisfies them. For example, in the first equation:\[a_1 \left(\frac{x_0 + x_1}{2}\right) + b_1 \left(\frac{y_0 + y_1}{2}\right) + c_1 \left(\frac{z_0 + z_1}{2}\right)\]Simplify this expression. Because both \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\) are solutions, their weighted average will satisfy the equation due to the linearity:\[a_1 x_0 + b_1 y_0 + c_1 z_0 = d_1\quad \text{and}\quad a_1 x_1 + b_1 y_1 + c_1 z_1 = d_1.\]

Step 4: Show Linearity Implication

Essentially, this step confirms by linearity that substituting the midpoint computes as follows:\[\frac{a_1 x_0 + a_1 x_1 + b_1 y_0 + b_1 y_1 + c_1 z_0 + c_1 z_1}{2} = d_1\]Since both halves equal \(d_1\) (and similar for \(d_2, d_3\)), the expression is true, showing that the midpoint satisfies the equation. This confirms that the system is consistent for the midpoint as well.

Step 5: Deduce Infinitely Many Solutions

Given that \(\left(\frac{x_0 + x_1}{2}, \frac{y_0 + y_1}{2}, \frac{z_0 + z_1}{2}\right)\) is a solution, observe that for any \(t\) in [0, 1], the weighted average: \[(t x_0 + (1-t) x_1, t y_0 + (1-t) y_1, t z_0 + (1-t) z_1)\]is also a solution. This means there are infinitely many solutions along the line defined by the two points.

Key Concepts

Solution of Linear Equations

Linear equations play a crucial role in many areas of mathematics and other disciplines such as physics and engineering. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. They are typically expressed in the form: \[\ ax+b=c.\]
In a system of linear equations, you have multiple equations that need to be solved simultaneously. The key to finding the solution is determining the set of values that satisfy all of the equations in the system at once.
Different methods exist for solving these systems, such as substitution, elimination, and using matrices. In this particular problem, we see that solutions also have a geometric interpretation: they can be seen as points in space satisfying the mentioned equations. Furthermore, when dealing with solutions, determining whether they are unique, non-existent or infinite is often as instructional as finding a solution.

Systems of Equations

A system of equations consists of two or more equations with the same set of unknowns. Systems of equations can be classified based on the number of solutions they may have:

• Consistent and independent: A single solution exists. It is independent because no redundancy exists in the equations.
• Consistent and dependent: There are infinitely many solutions, indicating the equations describe the same plane or line in space.
• Inconsistent: No solution exists because the equations describe parallel planes or lines that never intersect.

When dealing with linear systems, a geometric view can be helpful. For instance, in three dimensions, each equation represents a plane, and solutions are points where these planes intersect. Solving a system can therefore be understood as finding the intersections of these planes.

Infinite Solutions

Infinite solutions in a system of linear equations occur when the equations describe the same geometrical object, such as a line. In the original exercise, the given linear system has potentially infinite solutions because the individual equations do not intersect at a single point but along a line.
Discovering infinite solutions brings us to understand consistency and dependency in the system.The exercise shows that if two solutions exist like points \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\), then any point along the line connecting them, given by \((tx_0 + (1-t)x_1, ty_0 + (1-t)y_1, tz_0 + (1-t)z_1)\) for \(0 \leq t \leq 1\), is also a solution.
This is because linear equations retain consistency over lines formed by weighted combinations of solutions. Thus, demonstrating that if two solutions exist, an entire continuum of solutions becomes viable.