Problem 27 - Precalculus Mathematics for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered pair form given in Example 6. $$\left\\{\begin{array}{l}-x+y=2 \\\4 x-3 y=-3\end{array}\right.$$

Step-by-Step Solution

Answer

The system has a unique solution: (3, 5).

Step 1: Choose a variable to eliminate

We'll eliminate one of the variables by combining both equations. Let's choose to eliminate \(y\) by aligning the coefficients of \(y\) in both equations.

Step 2: Adjust equations for elimination

To eliminate \(y\), we can multiply the first equation \(-x + y = 2\) by 3, so the coefficient of \(y\) becomes 3, to match the \(-3\) in the second equation.Equation 1 becomes:\[-3x + 3y = 6\]

Step 3: Add the adjusted equations

Add the modified first equation to the second equation to eliminate \(y\):\[\begin{align*}-3x + 3y &= 6 \4x - 3y &= -3 \\hlinex &= 3\end{align*}\]

Step 4: Substitute to find the other variable

Substitute \(x = 3\) back into the original first equation to solve for \(y\):\[-x + y = 2\]\[-3 + y = 2\]\[y = 5\]

Step 5: Combine the solution

With \(x = 3\) and \(y = 5\), the solution to the system is \((3, 5)\).

Key Concepts

Elimination Method

The Elimination Method is a powerful technique for solving systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other.

To effectively use this method, follow these general steps:

• Choose which variable you want to eliminate from the equations. This choice can depend on the coefficients or personal preference.
• Adjust the equations, if necessary, by multiplying one or both of them so that the coefficients of the chosen variable are equal in magnitude but opposite in sign.
• Add or subtract the equations to eliminate the chosen variable.
• Solve the resulting single-variable equation.
• Substitute back to find the value of the other variable.

The beauty of the Elimination Method lies in its systematic approach, often leading to an ordered pair that represents the solution to the original system of equations.

Linear Equations

Linear Equations are the foundation of algebra and an essential mathematical concept. A linear equation is any equation that, when graphed, produces a straight line. These equations can have one or more variables, but each variable is raised to the first power only.

In the context of a system of linear equations, you'll likely see them expressed in a form such as \(-x + y = 2\) or \(4x - 3y = -3\). Here, some key features stand out:

• The expression involves only variables and constants.
• Each term in the equation is either a constant or the product of a constant and a variable.
• The graph of a linear equation is a straight line.
• These equations often represent real-world situations, like speed and distance problems.

Understanding how to work with and manipulate linear equations is crucial. They form the basis for more complex algebraic processes and problem-solving techniques according to the scenario you're working with.

Ordered Pair Solutions

Ordered Pair Solutions are a way of representing the solutions to a system of equations. An ordered pair consists of two elements, usually written in the form \(x, y\). These pairs signify the point of intersection of the equations when graphed, indicating the values of the variables that satisfy both equations.

In our example, the ordered pair solution \(3, 5\) means that when \(x = 3\) and \(y = 5\), both linear equations are true. Here’s why ordered pairs are important:

• They provide a concise and clear way to display the solution.
• They show the intersection point on a graph, which can be critical for visualization.
• Ordered pairs are universally recognisable, facilitating common understanding in mathematics.

Ordered pairs are not just a mathematical notation but a powerful tool for verifying that a particular set of values satisfies a given system of equations.