Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by elimination. $$\begin{array}{l}x+y=5 \\\x-y=1\end{array}$$

Step-by-Step Solution

Answer

The solution is \( (3, 2) \).

Step 1: Align the Equations

Write down the system of equations so they are visually aligned to recognize like terms:\[\begin{align*}x + y &= 5 \x - y &= 1\end{align*}\]

Step 2: Eliminate the Variable y

Add the two equations together. This will eliminate the variable \(y\) because \(+y\) and \(-y\) cancel each other out:\[\begin{align*}(x + y) + (x - y) &= 5 + 1 \2x &= 6\end{align*}\]

Step 3: Solve for x

Divide both sides of the equation by 2 to solve for \(x\):\[\begin{align*}x &= \frac{6}{2} \x &= 3\end{align*}\]

Step 4: Substitute x into an Original Equation

Use the value of \(x\) to find \(y\) by substituting \(x = 3\) back into one of the original equations, for example, \(x + y = 5\):\[\begin{align*}3 + y &= 5\end{align*}\]

Step 5: Solve for y

Subtract 3 from both sides to solve for \(y\):\[\begin{align*}y &= 5 - 3 \y &= 2\end{align*}\]

Step 6: Verify the Solution

Check the solution \((x, y) = (3, 2)\) by substituting back into both original equations:1. \(3 + 2 = 5\), which is true.2. \(3 - 2 = 1\), which is also true. This confirms the solution is correct.

Key Concepts

Understanding System of Equations

A system of equations is a collection of two or more equations with a common set of variables. The goal when working with a system of equations is to find the values of these variables that satisfy all the equations simultaneously. In our example, we have two equations:

• \( x + y = 5 \)
• \( x - y = 1 \)

These equations share the same variables, \(x\) and \(y\), indicating that both equations are interconnected. Solving systems of equations allows us to find a point where both equations meet, often depicted graphically as the intersection points of lines or curves. Systems of equations appear frequently in real-world situations when you need to find relationships between two changing quantities. Breaking them down into manageable parts aids in understanding the dynamics of the relationship.

Steps to Solving Equations

Solving equations involves finding the value(s) of unknown variables that satisfy the equation(s). When dealing with systems of equations, several methods can be employed, such as graphing, substitution, or elimination. The elimination method is often preferred in cases where the coefficients of one of the variables are opposites, making it easy to cancel out one variable by adding or subtracting equations.
In the provided example, the elimination method is used. We start by writing the system of equations vertically to easily compare and align like terms. This visual arrangement helps in identifying how to combine the equations effectively. By carefully adding the two equations together, the \( y \) terms cancel each other out, simplifying the process to find \( x \). After determining the value of \( x \), we substitute it back into one of the original equations to find \( y \). The final step is a double-check to ensure that the found values satisfy all given equations, confirming the solution's accuracy.

The Art of Variable Elimination

Variable elimination is a powerful algebraic technique used to reduce the number of unknowns in a system, simplifying the system to make it more manageable. The essence of this method is to strategically add or subtract equations to cancel out one of the variables. In the given problem:

• We start with two equations: \(x + y = 5\) and \(x - y = 1\).
• The objective is to eliminate \(y\) by adding them: \((x + y) + (x - y) = 5 + 1\).

Upon addition, the \(+y\) and \(-y\) cancel each other, leaving us with \(2x = 6\). This brings us to a much simpler equation with just one variable. Solving this single-variable equation is straightforward. After finding \(x\), the value is substituted back into any of the original equations to find the value of \(y\).
Variable elimination is especially useful in solving linear systems where direct combination of equations leads to quick results. It streamlines the problem-solving process, making it not only efficient but also intuitive once the basic principle of cancelling out variables is grasped.