Problem 12 - Precalculus with Limits

Question

Solving a System by Elimination In Exercises \(5-12,\) solve the system by the method of elimination. Label each line with its equation. To print an enlarged copy of the graph, go to MathGraphs.com. $$ \left\\{\begin{aligned} 9 x-3 y &=-15 \\\\-3 x+y &=5 \end{aligned}\right. $$

Step-by-Step Solution

Answer

This system of equations has infinite solutions, represented by any points (x, y) which satisfy the equation \(y = 3x + 5\).

Step 1: Multiply the Equations

In order to eliminate one variable by adding or subtracting the equations, it's best to have the coefficients before the y's in both equations to be equal (ignoring the sign). Consider the equations. The second equation is multiplied by 3 to match the coefficient of y in equation 1, thus yielding \(9x-3y = -15\) and \(-9x +3y = 15\).

Step 2: Add/Subtract the Equations

Add the two equations together. This results in \(0x + 0y = 0\), or simply \(0 = 0\). This result implies that the two original equations are dependent and represent the same line. Therefore, they have infinite points in common or in other words, there are infinite solutions.

Step 3: Write the Infinite Solutions

In this case, the solution to the system of equations is the equation of the line they represent. Thus, the solution could be expressed as any arbitrary pair (x, y) satisfying either (they are the same) equation. For instance, using the second equation: \(y = 3x + 5\), any point (x, y) satisfying this equation is a solution to the system.

Key Concepts

Elimination Method

The elimination method is a strategic way to solve systems of linear equations. When using the elimination method, the goal is to eliminate one variable so that you can easily solve for the other. This is achieved by manipulating the equations through multiplication, addition, or subtraction, until the coefficients of one variable cancel each other out when the equations are added together.

For example, let's say you have two equations: Equation 1 (9x - 3y = -15) and Equation 2 (-3x + y = 5). In order to eliminate one of the variables, you can multiply Equation 2 by 3, which then becomes (-9x + 3y = 15). Now, adding this to Equation 1 would eliminate the 'y' variable since -3y and +3y add up to 0. This method can quickly lead to a solution when done correctly. In cases where the result is an identity like 0 = 0, it gives us special insight into the nature of the system.

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. These equations represent straight lines when graphed on a coordinate plane. The solution to the system of equations is the point or points where the lines intersect.

The process of finding this point of intersection can be approached in various ways, including graphing, substitution, and elimination. Systems can have one unique solution (an intersection at a single point), no solution (if the lines are parallel and never intersect), or infinitely many solutions (if the lines coincide). The elimination method is particularly useful when the equations are set up in such a way that one variable can be easily removed, as we see with our example exercise.

Dependent System

In the context of linear equations, a dependent system arises when all the equations in the system describe the same line. This means that rather than intersecting at a single point, the lines lie on top of each other, resulting in an infinite number of intersections – that is, infinite solutions.

In our exercise, after using the elimination method, we get the equation 0 = 0, which confirms that the system is dependent. This identity is true regardless of the values of 'x' and 'y,' which means that every point on the line will satisfy both equations. To visualize a dependent system, picture a single line on a graph with countless points; each one is a solution to the system.

Infinite Solutions

Infinite solutions occur when a system of linear equations has a countless number of solutions. This typically happens in a dependent system, where the two equations represent the same exact line. Since every point on the line satisfies both equations, there are infinite pairs of (x, y) that solve the system.

In practice, when we obtain an equation like 0 = 0 after eliminating variables, it indicates infinite solutions. From our step-by-step exercise, once we determined the system is dependent, any point that lies on the line represented by the equations is a solution. For instance, using the second equation, y = 3x + 5, every single point that satisfies this equation is part of the infinite set of solutions for the system.