Problem 7 - Algebra and Trigonometry

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 3. $$\left\\{\begin{aligned} x+y &=4 \\\\-x+y &=0 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \((2, 2)\); the system has one solution.

Step 1: Analyze the Equations

We are given the system of equations: \(x + y = 4\) and \(-x + y = 0\). Notice that both equations are in a standard linear form: \(ax + by = c\).

Step 2: Add the Equations

Add the two equations together to eliminate \(x\): \(x + y = 4\) plus \(-x + y = 0\) results in \(2y = 4\).

Step 3: Solve for y

Divide both sides of \(2y = 4\) by 2 to isolate \(y\): \(y = 2\).

Step 4: Substitute to Find x

Substitute \(y = 2\) back into the first equation \(x + y = 4\): \(x + 2 = 4\). Subtract 2 from both sides to solve for \(x\): \(x = 2\).

Step 5: Verify the Solution

Substitute \(x = 2\) and \(y = 2\) into the second equation: \(-x + y = 0\) becomes \(-2 + 2 = 0\), which holds true. Both equations are satisfied by \((x, y) = (2, 2)\).

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share the same set of variables. In this exercise, we are working with a system involving two variables, \(x\) and \(y\). Each equation in the system represents a linear relationship between these variables. The goal is to find a common solution that satisfies all equations simultaneously.

Systems of equations come in three varieties:

• Independent systems: These have a single unique solution where the lines intersect at one point.
• Dependent systems: These have infinitely many solutions because the lines are coincident, meaning they overlap completely.
• Inconsistent systems: These have no solution as the lines are parallel and do not intersect.

In our given system, we see the system is independent, leading us to one unique solution. Understanding the nature of the system is crucial as it guides how we approach solving it.

Solving Linear Equations

Solving linear equations involves finding the values of variables that satisfy the equation. Linear equations are equations where the highest power of the variable is 1.

To solve these equations, there are basic steps you can follow:

• Use addition, subtraction, multiplication, or division to isolate the variable on one side of the equation.
• Simplify both sides to solve for the variable.
• Verify your solution by substituting it back into the original equation.

In the context of systems of equations, solving each linear equation helps in finding the intersection point. In the example provided, once we found \(y = 2\), we substituted it back to find \(x = 2\). Each step should simplify the equations until the solution is unmistakably clear.

Elimination Method

The elimination method is a technique used to solve systems of linear equations. It involves combining the equations in a way that eliminates one of the variables, making it easier to solve for the other.
This method is especially useful when the coefficients of one variable in the two equations are opposites.

Here's how you can apply the elimination method:

• Add or subtract the equations to eliminate one of the variables.
• Solve for the remaining variable after the elimination.
• Substitute back into one of the original equations to find the value of the eliminated variable.

For our example, the addition of the two equations, \(x + y = 4\) and \(-x + y = 0\), perfectly eliminated \(x\) leading us to quickly solve for \(y=2\). Using elimination can simplify complex systems efficiently, allowing for faster solutions.