Question

Find the solution to the given linear system. If the system has infinite solutions, give 2 particular solutions. $$ \begin{aligned} x_{1}+x_{2} &=3 \\ 2 x_{1}+x_{2} &=4 \end{aligned} $$

Short answer

The system has a unique solution: \((x_1, x_2) = (1, 2)\).

Step-by-step solution

Write the System of Equations

We are given the linear system: \( \begin{aligned} x_1 + x_2 &= 3 \ 2x_1 + x_2 &= 4 \end{aligned} \). We will solve for \(x_1\) and \(x_2\).

Use Substitution or Elimination

To solve the system, we can use either substitution or elimination. Here, we will use the elimination method to eliminate \(x_2\) from one of the equations.

Subtract Equations

Subtract the first equation from the second to eliminate \(x_2\): \( (2x_1 + x_2) - (x_1 + x_2) = 4 - 3 \). Simplifying gives \( x_1 = 1 \).

Substitute Back to Find Other Variable

Use \(x_1 = 1\) in the first equation to find \(x_2\). Substitute into \(x_1 + x_2 = 3\), so \(1 + x_2 = 3\). Solving for \(x_2\) gives \(x_2 = 2\).

Solution Statement

The solution to the system is \((x_1, x_2) = (1, 2)\), a unique solution.

Key concepts

Substitution Method

The substitution method is an effective way to solve linear systems of equations. It involves solving one of the equations for a single variable and then substituting this expression into the other equations. This helps to reduce the system to a single equation in one variable. Here's a simple walkthrough of how it works:

• Solve one equation for one variable: Pick one of the equations and solve for one of the variables. For example, solve for \(x_1\) in the first equation: \(x_1 = 3 - x_2\).
• Substitute into the second equation: Take the expression for \(x_1\) and substitute it into the second equation: \(2(3 - x_2) + x_2 = 4\).
• Solve the resulting single-variable equation: Now, solve for \(x_2\) and then use this value to find \(x_1\).

This method is particularly helpful when one of the equations is easy to solve for one of the variables. It simplifies the problem by focusing on one variable at a time.

Elimination Method

The elimination method is another powerful approach to solve linear systems of equations. It's designed to eliminate one of the variables by adding or subtracting the equations, thus reducing the system to a single-variable equation. Let's see how it works:

• Align the equations: Look at both equations to see if the coefficients are aligned for elimination. In our example, the coefficients of \(x_2\) are the same.
• Add or subtract the equations: By subtracting the first equation from the second, \((2x_1 + x_2) - (x_1 + x_2) = 4 - 3\), we eliminate \(x_2\).
• Solve for the remaining variable: Solving this gives \(x_1 = 1\).
• Substitute back: Use \(x_1 = 1\) in any equation to find \(x_2\). Inserting into \(x_1 + x_2 = 3\), we find \(x_2 = 2\).

The elimination method is best used when the coefficients of one variable are already aligned, or can be easily adjusted to make terms cancel out.

Unique Solution

A unique solution in a linear system means that there is only one set of values that satisfies all equations in the system. This occurs when the lines represented by the equations intersect at a single point. Let's explore why our system has a unique solution:

• Independent Equations: In our example, both equations represent two different lines with distinct slopes. When the lines are not parallel, they will intersect at one point.
• Solving Confirms Uniqueness: Solving the system through elimination or substitution gives us a consistent, single answer: \((x_1, x_2) = (1, 2)\).
• No Infinite Solutions or Contradictions: Since the solution is consistent, there are no contradictions (which would suggest no solutions) and no infinite number of solutions (as would happen if the lines were identical or overlapping).

Thus, a unique solution arises when two linear equations intersect at exactly one point, confirming that our calculated solution is accurate and reliable.