Problem 3 - Differential Equations and Linear Algebra

Question

Verify that for all values of \(t\) $$(1-t, 2+3 t, 3-2 t)$$ is a solution to the linear system $$\begin{aligned}x_{1}+x_{2}+x_{3} &=6. \\\x_{1}-x_{2}-2 x_{3} &=-7. \\\5 x_{1}+x_{2}-x_{3} &=4.\end{aligned}$$

Step-by-Step Solution

Answer

The given vector \((1-t, 2+3t, 3-2t)\) is a solution to the linear system for all values of \(t\) as the simplified equations \(6 + 0t = 6\), \(-7 - 2t + 4t = -7\), and \(4 + 0t = 4\) are true for any value of \(t\).

Step 1: Substitute given vector into equations

Replace \(x_1\) with \(1-t\), \(x_2\) with \(2+3t\), and \(x_3\) with \(3-2t\) in the given linear system of equations. We get the following modified equations: 1. \((1-t) + (2+3t) + (3-2t) = 6\) 2. \((1-t) - (2+3t) - 2(3-2t) = -7\) 3. \(5(1-t) + (2+3t) - (3-2t) = 4\)

Step 2: Simplify and analyze equations

Simplify the modified equations: 1. \(1 - t + 2 + 3t + 3 - 2t = 6\) 2. \(1 - t - 2 - 3t - 6 + 4t = -7\) 3. \(5 - 5t + 2 + 3t - 3 + 2t = 4\) Combine like terms and check if the equations are true for any value of \(t\): 1. \(6 + 0t = 6\) 2. \(-7 - 2t + 4t = -7\) 3. \(4 + 0t = 4\)

Step 3: Conclusion

As we can see, all three simplified equations are true for any value of \(t\), without having \(t\) in the equations. Thus, we can conclude that \((1-t, 2+3t, 3-2t)\) is a solution to the given linear system for all values of \(t\).

Key Concepts

Solution Verification

When solving a system of linear equations, checking the validity of an assumed solution is essential. This process is known as solution verification. In this exercise, we're given the parametric solution \((1-t, 2+3t, 3-2t)\) and asked to verify it against the system of equations:

• \(x_1 + x_2 + x_3 = 6\)
• \(x_1 - x_2 - 2x_3 = -7\)
• \(5x_1 + x_2 - x_3 = 4\)

The process involves substituting the given expressions for \(x_1\), \(x_2\), and \(x_3\) into each equation individually. After substitution, we simplify each equation to see if it remains true for all values of \(t\). This involves combining like terms and checking if the equation resolves correctly.For example, take the first equation:

• After substitution: \,\((1-t) + (2+3t) + (3-2t) = 6\)
• After simplifying: \,\(6 + 0t = 6\)

This shows that the equation holds true. By verifying each equation individually, we confirm that the solution is valid for all values of \(t\).

Parametric Solution

A parametric solution is a way to express the variables in a system of equations in terms of one or more parameters. These parameters can represent any value within a particular set, often giving the solution more flexibility.In this exercise, the parametric solution is given in terms of \(t\) as \((1-t, 2+3t, 3-2t)\). This means for each value of \(t\), the solution provides values for \(x_1\), \(x_2\), and \(x_3\). By having a parametric representation, we address infinitely many specific solutions simultaneously rather than finding solutions point by point.Think of \(t\) as a slider that can adjust the values of \(x_1\), \(x_2\), and \(x_3\) while still satisfying the original equations. This is particularly powerful in linear algebra as it allows for comprehensive understanding of the solution space of the system. It highlights scenarios where either multiple intersections occur or the entire plane of solutions fits the system.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find the values for these variables that satisfy all equations simultaneously. In this exercise, we have a system consisting of three equations with three variables (\(x_1\), \(x_2\), and \(x_3\)):

• \(x_1 + x_2 + x_3 = 6\)
• \(x_1 - x_2 - 2x_3 = -7\)
• \(5x_1 + x_2 - x_3 = 4\)

Such systems might have no solution, a unique solution, or infinitely many solutions, which depends on the nature of the equations. A common method to solve these systems is to use substitution, elimination, or matrix methods like row reduction.In this case, the given parametric solution \((1-t, 2+3t, 3-2t)\) provides a general solution indicating that there are infinitely many solutions, as the system holds true for every value of \(t\). Understanding how systems of linear equations can be manipulated and solved helps us explore not only individual solutions but also the span and dimension of solution sets in a vector space.