Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by elimination. $$\begin{array}{c}-4 x-2 y=-2 \\\6 x+y=7\end{array}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{2}\) and \(y = -2\).

Step 1: Align the Equations

Write down the given system of equations in a standard form, and ensure that both equations are aligned:\(-4x - 2y = -2\)\(6x + y = 7\).

Step 2: Eliminate One Variable

To eliminate one variable, we can add the two equations after adjusting them so that the coefficients of either \(x\) or \(y\) will cancel each other out. Multiply the second equation by 2 to eliminate \(y\):\(-4x - 2y = -2\)\(12x + 2y = 14\).

Step 3: Add the Equations

Add the equations after making the adjustment:\((-4x - 2y) + (12x + 2y) = -2 + 14\).This simplifies to \(8x = 12\).

Step 4: Solve for x

Solve the equation obtained from the previous step for \(x\):Divide both sides by 8:\(x = \frac{12}{8}\).Simplify the fraction:\(x = \frac{3}{2}\).

Step 5: Substitute to Find y

Substitute \(x = \frac{3}{2}\) back into one of the original equations to find \(y\). We'll use the second original equation:\(6x + y = 7\).Substitute \(x = \frac{3}{2}\):\(6(\frac{3}{2}) + y = 7\).Simplify:\(9 + y = 7\).Solve for \(y\):\(y = 7 - 9 = -2\).

Step 6: Conclusion

The solution to the system of equations is \(x = \frac{3}{2}\) and \(y = -2\).

Key Concepts

Elimination Method

The elimination method is a powerful technique used to solve systems of linear equations. This approach focuses on removing or eliminating one of the variables by combining the equations, making it easier to solve for the remaining unknowns. The goal is to cancel out one variable by ensuring that its coefficients add up to zero when the equations are combined. This usually involves tweaking the equations (e.g., through multiplication) before adding or subtracting them to cancel out a variable.

The key steps include:

• Aligning the equations properly so their corresponding variables stack up for simpler arithmetic operations.
• Multiplying one or both of the equations by a suitable factor to make the coefficients of one variable opposites.
• Adding or subtracting the equations to eliminate a variable.

Choosing the elimination method generally depends on variable alignment and the ease of manipulating coefficients. It is efficient and widely used, especially when at least one of the variables can be easily canceled.

Solving Linear Equations

Solving linear equations is at the heart of finding solutions in a system of equations. Once we've successfully eliminated one of the variables, the next step is straightforward: solve the simplified equation for the remaining variable. Linear equations adhere to a standard format: variables and constants on one side, and the result on the other. For any linear equation of form \(ax + b = c\), we can isolate \(x\) by performing inverse operations:

• Subtract \(b\) from both sides if it exists on the constant side.
• Divide by the coefficient \(a\) to solve for the variable.

This step often follows quickly from the elimination process, as seen in the exercise where solving \(8x = 12\) was achieved by dividing both sides by 8. Finding one variable unlocks the ability to substitute and solve for the other, unraveling the system's complete solution.

Alignment of Equations

Equation alignment is a critical preparatory step in the elimination method. Proper alignment ensures the parts of the equations that represent the same variables are vertically stacked. This simplifies any additions or subtractions, keeping calculations organized and reducing errors.

When aligning, focus on:

• Standardizing equations to have variables on one side and constants on the other. This often requires simple rearrangements or restating the equations.
• Ensuring that the coefficients line up, making it easier to manipulate them for elimination.

In the original problem, aligning the equations meant writing them neatly together as \(-4x - 2y = -2\) and \(6x + y = 7\). This setup paved the way for selecting which variable to focus on for elimination by evaluating and potentially adjusting coefficients. Aligning correctly is a step that sets the foundation for solving the system efficiently and effectively.