Problem 12 - Elementary Algebra

Question

Use the elimination-by-addition method to solve each system. $$\left(\begin{array}{l}5 x-y=6 \\ 10 x-2 y=12\end{array}\right)$$

Step-by-Step Solution

Answer

Infinitely many solutions exist, represented by \(y = 5x - 6\).

Step 1: Analyze the System of Equations

The system has two equations: 1) \(5x - y = 6\) 2) \(10x - 2y = 12\)Notice that the second equation is a multiple of the first equation. Specifically, each term in the first equation is doubled to obtain the second equation.

Step 2: Inspect the Coefficients

If you multiply the first equation by 2, you get the second equation: 1) \(2(5x - y = 6)\) gives \(10x - 2y = 12\) Since the equations are dependent, this system has infinitely many solutions.

Step 3: Solve for y in Terms of x

Using the first equation, solve for \(y\) in terms of \(x\):\(5x - y = 6 \rightarrow y = 5x - 6\). This represents one equation for the solutions.

Step 4: Describe the Solution Set

The solution set is given by all pairs \((x, y)\) such that \(y = 5x - 6\). This is a line, implying infinitely many solutions.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The main goal when dealing with these systems is to find the values of the variables that satisfy all the equations at the same time.
For example, the system of equations given here is:

• \(5x - y = 6\)
• \(10x - 2y = 12\)

Each equation represents a line on a graph. The solution to the system is any point or points where the lines intersect. To find these solutions, we can use various methods such as substitution, elimination by addition or subtraction, and graphing. In this particular exercise, we're using the elimination-by-addition method to simplify and solve the system.

Dependent Equations

Dependent equations occur when one equation in a system is a multiple of another. This means the equations essentially represent the same line.
Look at the system provided:

• The first equation is \(5x - y = 6\).
• The second equation \(10x - 2y = 12\) is exactly twice the first equation.

To see this relationship, multiply the first equation by 2, turning it into the second equation. This shows the lines overlap completely, indicating that they're dependent. Dependent equations are a telltale sign of a special type of solution set, which we'll discuss next.

Infinitely Many Solutions

When you have dependent equations like in this system, it implies there are infinitely many solutions. This means any point on one line is also a point on the other. The two lines are essentially the same.
In our system:

• When we manipulate the first equation, we express \(y = 5x - 6\).
• All pairs \((x, y)\) that satisfy this equation are solutions to the system.

Since the lines overlap, every value of \(x\) will have a corresponding \(y\) that satisfies the equation \(y = 5x - 6\).
This full line of solutions indicates the nature of dependent equations: a consistent system of lines sharing all points.