Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by elimination. $$\begin{aligned}&3 x+y=6\\\&6 x+2 y=1\end{aligned}$$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Understand the Equations

We are given two equations: \(3x + y = 6\) and \(6x + 2y = 1\). The goal is to eliminate one variable by combining the equations.

Step 2: Multiply to Align Coefficients

Notice that the second equation \(6x + 2y = 1\) is exactly double the first equation \(3x + y = 6\). Let's multiply the first equation by 2 to align the coefficients of \(x\):\[2(3x + y) = 2(6)\]Which gives us: \(6x + 2y = 12\).

Step 3: Subtract the Equations

Subtract the new equation \(6x + 2y = 12\) from the original second equation \(6x + 2y = 1\):\[(6x + 2y) - (6x + 2y) = 1 - 12\]This simplifies to: \(0 = -11\).

Step 4: Analyze the Result

The equation \(0 = -11\) is a contradiction, which means there is no possible solution that can satisfy both equations simultaneously. This implies the system of equations has no solution.

Key Concepts

Elimination Method

The elimination method is a technique used to solve systems of linear equations by removing one of the variables. This makes the system easier to solve. To use this method, we manipulate the given equations such that adding or subtracting them eliminates one variable.

• Start with aligning the coefficients of one variable in the two equations. This usually requires multiplying one or both equations by a constant.
• Once the coefficients match, we either add or subtract the equations to eliminate the targeted variable.
• The resulting equation can then be solved to find the value of the remaining variable.

Remember, the goal is to manipulate the equations until one of the variables disappears. In our exercise, after aligning the coefficients, subtracting the equations led to an impossible result, which helps us conclude the problem characteristics.

No Solution

In some systems of linear equations, you'll find that they have no solution. This occurs when the equations represent parallel lines that never intersect. In mathematical terms, it often results in an impossible statement like the one in our example, where we derived the equation \(0 = -11\), which is not true.

• When you get a statement like \(0 = a\), where \(a\) is any non-zero number, it indicates no solution.
• This often happens when two equations are essentially multiples of the same equation but have different constants on the right side.
• Such equations describe parallel lines without any point of intersection, thus confirming no solution to the system.

Understanding when there's no solution can save time and help detect inconsistencies between the equations early in the solving process.

Linear Equations

Linear equations are fundamental in algebra and consist of equations involving variables raised to the power of one. They represent straight lines when graphed on a coordinate plane. A system of linear equations involves finding the point(s) where these lines intersect.

• Each linear equation can be expressed in the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
• When solving a system of linear equations, we're seeking the values of \(x\) and \(y\) that satisfy all equations simultaneously.
• There are generally three types of solutions to a system:
  • A single solution (lines intersect at one point)
  • Infinitely many solutions (lines are identical)
  • No solution (lines are parallel)

Linear equations are prevalent across different fields, including physics and economics, where they're used to model real-world situations. Understanding their behavior through graphing and algebraic methods is a key part of solving these systems effectively.