Problem 21 - 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. $$\left\\{\begin{array}{r} x+y=4 \\ -x+y=0 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has one solution: \((2, 2)\).

Step 1: Understanding the System

We have two linear equations that form a system: \( x + y = 4 \) and \( -x + y = 0 \). The goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.

Step 2: Solving the Second Equation for x

Let's solve the second equation \( -x + y = 0 \) for \( x \). We can rearrange it as \( x = y \). This expresses \( x \) in terms of \( y \).

Step 3: Substituting x into the First Equation

Substitute \( x = y \) into the first equation \( x + y = 4 \). This gives us \( y + y = 4 \) or \( 2y = 4 \).

Step 4: Solving for y

Divide both sides of \( 2y = 4 \) by 2 to solve for \( y \). This yields \( y = 2 \).

Step 5: Finding the Value of x

Now that we know \( y = 2 \), we can use \( x = y \) to find \( x \). Therefore, \( x = 2 \) as well.

Step 6: Write the Solution as an Ordered Pair

The solution to the system is expressed as the ordered pair \( (x,y) = (2, 2) \). This is the only solution, meaning there is not an infinite number of solutions, and the system is consistent with one unique solution.

Key Concepts

Linear Equations

Linear equations are expressions of two variables that show a relationship in which each equation is a straight line when graphed on a coordinate plane. In our given system, we have the linear equations \( x + y = 4 \) and \( -x + y = 0 \). Each equation represents a line in the 2D space, defined by their respective "slope-intercept" forms. These forms show how one variable depends on the other, and how they interact.Understanding linear equations involves knowing that they can have different forms, such as:

• Standard form: \( ax + by = c \)
• Slope-intercept form: \( y = mx + b \) where \( m \) is the slope and \( b \) is the intercept.
• Point-slope form: \( y - y_1 = m(x - x_1) \)

The essential idea is to find solutions (i.e., the values of \(x\) and \(y\)) that satisfy both equations simultaneously, which means finding points where both lines intersect on the graph. Our linear system solution tells us that the point of intersection is where the values of \(x\) and \(y\) fulfill both equations.

Ordered Pair Solution

An ordered pair is a set of numbers used to locate a point on a coordinate grid or plane. It is commonly written in the form \((x, y)\), where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate. In solving systems of equations, an ordered pair represents the point where two lines intersect.In our exercise, the ordered pair solution for the system is \((2, 2)\). This means that both the equations \( x + y = 4 \) and \( -x + y = 0 \) are satisfied when \( x = 2 \) and \( y = 2 \).
The process of obtaining this solution involved:

• Substituting one variable's expression in terms of another into one of the equations.
• Solving the resulting equation to find one variable's value.
• Using this value in the other equation to find the second variable.

Knowing how to express solutions as ordered pairs is crucial in graphing, where visual representation can provide deeper insights into the nature of the solution set.

Unique Solution

A system of linear equations might have a unique solution, infinitely many solutions, or no solution at all. The unique solution occurs when there is exactly one set of values for the variables \(x\) and \(y\) that satisfies both equations.In this particular exercise, the system has a unique solution: the ordered pair \((2, 2)\). Both lines intersect at this single point on the graph.
Understanding the nature of solutions involves:

• Intersections: When two lines intersect at exactly one point, the system is consistent with a unique solution.
• Parallel Lines: If the lines are parallel and do not intersect, the system has no solution.
• Coinciding Lines: When the lines overlap completely, every point on the line is a solution, leading to infinitely many solutions.

The concept of a unique solution is crucial in understanding systems of linear equations, as it implies that there is one and only one intersection point, ensuring a definitive answer to the system of equations.