Problem 49 - Precalculus: Mathematics for Calculus

Question

(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 $$ \left\\{\begin{array}{l} a_{1} x+b_{1} y+c_{1} z=d_{1} \\ a_{2} x+b_{2} y+c_{2} z=d_{2} \\ a_{3} x+b_{3} y+c_{3} z=d_{3} \end{array}\right. $$ Show that \(\left(\frac{x_{0}+x_{1}}{2}, \frac{y_{0}+y_{1}}{2}, \frac{z_{0}+z_{1}}{2}\right)\) is also a solution. (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

The midpoint of two solutions is also a solution, hence infinite points (solutions) on the line segment exist if two solutions are given.

Step 1: Understand the System of Equations

The system of linear equations is represented as follows:\[\begin{align*}a_{1} x + b_{1} y + c_{1} z &= d_{1}, \a_{2} x + b_{2} y + c_{2} z &= d_{2}, \a_{3} x + b_{3} y + c_{3} z &= d_{3}.\end{align*}\]We are given two solutions: \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\). We need to verify if their midpoint \(\left(\frac{x_{0}+x_{1}}{2}, \frac{y_{0}+y_{1}}{2}, \frac{z_{0}+z_{1}}{2}\right)\) is also a solution.

Step 2: Substitute the Midpoint into the Equations

Substitute \( x = \frac{x_0 + x_1}{2}, \; y = \frac{y_0 + y_1}{2}, \; z = \frac{z_0 + z_1}{2} \) into each equation in the system:For the first equation:\[a_1 \left(\frac{x_0 + x_1}{2}\right) + b_1 \left(\frac{y_0 + y_1}{2}\right) + c_1 \left(\frac{z_0 + z_1}{2}\right) = \frac{1}{2}(d_1 + d_1) = d_1.\]The calculations are similar for the other two equations, confirming the midpoint is a solution.

Step 3: Verify the Arithmetic

The substitution yields:\[a_1 \left(\frac{x_0 + x_1}{2}\right) = \frac{a_1 x_0 + a_1 x_1}{2}, \;b_1 \left(\frac{y_0 + y_1}{2}\right) = \frac{b_1 y_0 + b_1 y_1}{2}, \;c_1 \left(\frac{z_0 + z_1}{2}\right) = \frac{c_1 z_0 + c_1 z_1}{2}.\]Adding these proves: \[\frac{1}{2}(a_1 x_0 + b_1 y_0 + c_1 z_0) + \frac{1}{2}(a_1 x_1 + b_1 y_1 + c_1 z_1) = d_1.\] Identify analogous relations for the other equations.

Step 4: Conclude for Part (a)

Since each equation holds after substitution, \(\left(\frac{x_0 + x_1}{2}, \frac{y_0 + y_1}{2}, \frac{z_0 + z_1}{2}\right)\) is indeed a solution to the system. This confirms Part (a).

Step 5: Use Geometry to Assist Part (b)

If a line segment connecting two different solutions \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\) is made up entirely of solutions, then every point on this line is also a solution. Points in between can be described as: \[(x_t, y_t, z_t) = (1 - t)(x_0, y_0, z_0) + t(x_1, y_1, z_1)\]for \(t\) ranging from 0 to 1.

Step 6: Demonstrate Infinitely Many Solutions

Since for any \(t \in [0, 1]\), \((x_t, y_t, z_t)\) represents a solution, and \(t\) can take on any value in this interval, an infinite set of solutions exists in the form of a line segment, establishing that the system has infinitely many solutions. This concludes part (b).

Key Concepts

Solutions of Linear Equations

In a system of linear equations, multiple linear equations work together to find the values of variables such as \(x\), \(y\), and \(z\). Each equation in the system represents a line (in two dimensions) or a plane (in three dimensions). When we solve these equations, we are essentially looking for the point(s) where these lines or planes intersect. These intersection points are known as solutions.

• A system of linear equations may have a single solution, meaning all lines or planes intersect at one point.
• It can also have no solution if the lines or planes are parallel and never intersect.
• Furthermore, a system may also have infinite solutions, indicating all lines or planes coincide perfectly.

In these exercises, the solutions \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\) indicate that two distinct sets of values satisfy the entire system of equations. The important task here is to see whether more solutions exist when we know two have already been found.

Infinite Solutions

Infinite solutions arise when a system of linear equations has more than one solution running continually, like along a line or a plane.
Consider a scenario where two solutions, \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\), have already been identified. This suggests these two solutions form endpoints of a line segment composed entirely of other solutions. This line can be defined parametrically:
\[(x_t, y_t, z_t) = (1 - t)(x_0, y_0, z_0) + t(x_1, y_1, z_1)\]
Here, \(t\) is a parameter that ranges between 0 and 1. If we let \(t\) vary, every point on this line is a solution to the system.
In essence, this establishes the system has infinitely many solutions because altering \(t\) continuously fills the line. Parametric equations help visualize the continuous spectrum of solutions beyond isolated solutions.

Midpoint of Solutions

The midpoint of two points acts as a fundamental concept in analyzing and proving infinite solutions in linear systems. If we have two initial solutions, \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\), their midpoint is given by:
\[ \left( \frac{x_0 + x_1}{2}, \frac{y_0 + y_1}{2}, \frac{z_0 + z_1}{2} \right) \]
This mathematical operation is a straightforward average of each corresponding coordinate. Calculating this midpoint mathematically offers insight into whether a new solution lies between the two given solutions.
Substituting and checking these midpoint coordinates into each equation of the system allows us to verify their validity as solutions. Since we obtained the midpoint by averaging solutions, it must satisfy all original equations.
Thus, repeatedly using this midpoint method gives rise to a continuous sequence of solutions, helping illustrate why systems with two different solutions have infinitely many.