Problem 18 - Intermediate Algebra

Question

For Problems \(1-18\), use the elimination-by-addition method to solve each system. (Objective 1 ) $$ \left(\begin{array}{rr} 5 x-y=6 \\ 10 x-2 y=12 \end{array}\right) $$

Step-by-Step Solution

Answer

The system has infinitely many solutions, as the equations are dependent.

Step 1: Double-check the Alignment

Make sure the equations are aligned correctly with the variables and constants in place. The given system of equations is as follows:\[\begin{array}{c}5x - y = 6 \ 10x - 2y = 12\end{array}\]

Step 2: Identify the Coefficients for Elimination

Next, identify which coefficients can be manipulated to eliminate one variable. Notice that the coefficient of \(-y\) in the first equation is \(-1\) and in the second equation is \(-2\). These can be used to eliminate \(y\).

Step 3: Multiply to Align Coefficients for Elimination

Multiply the entire first equation by 2 so that the coefficients of \(y\) match in magnitude:\[\begin{array}{c}2(5x - y) = 2(6) \ 10x - 2y = 12\end{array}\]which simplifies to:\[10x - 2y = 12\]Now both equations are:\[\begin{array}{c}10x - 2y = 12 \ 10x - 2y = 12\end{array}\]

Step 4: Subtract the Equations

Subtract the new first equation from the second equation to eliminate both variables:\[(10x - 2y) - (10x - 2y) = 12 - 12\]This simplifies to:\[0 = 0\]which indicates that the original equations are dependent, implying an infinite number of solutions.

Step 5: Conclusion for Infinite Solutions

Since both equations are identical after simplification, our system has infinitely many solutions. Any \(x\) and \(y\) that satisfy one equation will satisfy the other.

Key Concepts

Systems of Equations

When talking about a system of equations, we're referring to a set of two or more equations that have common variables. These variables represent unknowns, and our goal is to find values that satisfy all the equations at the same time. In this particular exercise, we have a system with two equations:

• Equation 1: \(5x - y = 6\)
• Equation 2: \(10x - 2y = 12\)

To "solve" this system means to find every possible set of numbers \((x, y)\) that makes both equations true. Typically, systems of equations can be solved using methods such as substitution, elimination, or graphical methods. The elimination method is employed here, which is particularly useful when you want to eliminate one variable at the start to solve for the other more easily. This method involves aligning and manipulating the equations to cancel out one variable, making the system simpler to solve.

Infinite Solutions

Infinite solutions occur in a system of equations when the equations represent the same line in a graph. This means that every point on the line satisfies both equations, and there are countless possible \((x, y)\) pairs that work. When you use the elimination method, as in this exercise, you might find that after elimination, the result is a true statement like \(0 = 0\). This is a significant indicator.

We found:

• Both equations simplify down to \(10x - 2y = 12\).
• Subtracting one from the other gives \(0 = 0\).

This tells us that both equations are basically identical, just expressed differently. Therefore, they have the same infinite number of solutions. When graphed, each solution corresponds to a point along the same line shared by both equations.

Dependent Equations

Dependent equations arise in systems where one equation is essentially a multiple or a simple transformation of the other. This is a hallmark of infinite solutions, signaling that the equations describe the same relationship or the same line.

In our example:

• Equation 1 multiplied by 2 yields Equation 2.
• Both then equal \(10x - 2y = 12\).

This dependency implies that they don't provide any additional information apart from what one single line on the graph would offer. Consequently, any value for \(x\) that satisfies this relationship would determine a corresponding \(y\), each point thus representing the infinite solutions along this consistent line. Dependent equations mean the system behaves like a single equation because it doesn't impose more conditions beyond what is already described by the single line.