Problem 39 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

No solution, the system has no consistent solutions.

Step 1 - Identify the Multiplication Factor

Observe that both equations are linear and ready to be aligned for elimination, but the coefficients of either variable should match in magnitude for one of them. Notice that the coefficient of \(y\) in the first equation is \(1\) and in the second is \(2\), thus multiplying the first equation by \(2\) will create matching coefficients for \(y\).

Step 2 - Multiply the First Equation

Multiply every term in the first equation by \(2\):\[2(3x + y) = 2(6)\]This simplifies to:\[6x + 2y = 12\].

Step 3 - Subtract to Eliminate

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

Step 4 - Analyze the Result

The result \(0 = 11\) is a contradiction, which means there is no set of \((x, y)\) satisfying both equations simultaneously.

Key Concepts

Elimination Method

The elimination method is a technique used to solve systems of linear equations. This involves removing one of the variables by ensuring that its coefficient is equal in both equations, and then subtracting or adding the equations. To apply the elimination method, follow these steps:

• Identify target variable: Decide which variable you'd like to eliminate.
• Align coefficients: Ensure the coefficients for this variable are equal across both equations, often by multiplication.
• Subtract or add: Eliminate the target variable by subtracting or adding the equations.
• Solve the equation: Solve the resulting simpler equation for the remaining variable.

This method simplifies the process by reducing the system to a single equation with one variable, making it easier to find solutions.

Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. When graphed on a Cartesian plane, a linear equation forms a straight line.Linear equations can take several forms, such as:

• Standard form: \(ax + by = c\)
• Slope-intercept form: \(y = mx + b\)

In the context of systems of equations, solving using linear equations involves finding the values of variables that satisfy all equations in the system simultaneously. Linear equations are called so because of their graphical representation as straight lines. Thus, finding a common solution is equivalent to finding the intersection point(s) of these lines on the graph.

Contradiction in Systems

A contradiction in a system of equations arises when you obtain a statement that is obviously false, such as \(0 = 11\), from manipulating the equations. This indicates that there is no solution set that satisfies all equations simultaneously.Identifying a contradiction generally involves:

• Using methods like elimination or substitution to simplify the system.
• Arriving at an impossible statement through algebraic manipulation.

In real-world applications, a contradiction suggests that an assumed model or set of constraints cannot be true under any circumstances. Recognizing contradictions is crucial as it highlights that a different approach or assumption may be needed to solve a problem or model a situation accurately.