Problem 16 - Algebra A Combined Function

Question

Solve each system of equations by the addition method. If a system contains fractions or decimals, you may want to first clear each equation of fractions or decimals. $$ \left\\{\begin{array}{l} 2 x+y=6 \\ 4 x+2 y=12 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Identify Coefficients for Addition

Before adding the equations, we need to ensure that the coefficients of one of the variables are opposites so that they cancel each other. Here, the equations are:1. \(2x + y = 6\) 2. \(4x + 2y = 12\)Notice that the coefficients of \(y\) are already suitable for elimination since they can easily become opposites.

Step 2: Adjust Coefficients (Optional)

In this case, we don't need to adjust the coefficients because multiplying the first equation by \(-2\) will make the \(y\) coefficients equal and opposite:Multiply equation 1 by \(-2\):\[-2(2x + y) = -2(6)\]This transforms to:\(-4x - 2y = -12\).

Step 3: Add the Equations

Now, add the adjusted equation 1 to equation 2:Adjusted equation 1:\(-4x - 2y = -12\)Original equation 2:\(4x + 2y = 12\)Adding them results in:\(0x + 0y = 0\).This simplifies to \(0 = 0\), indicating dependent equations.

Step 4: Interpret the Result

The outcome \(0 = 0\) means the system has infinitely many solutions. This occurs when one equation is just a multiple of the other, showing that they are essentially the same line.

Key Concepts

Addition Method

The addition method is a strategic way to solve systems of linear equations. It's often referred to as the elimination method because it helps eliminate one of the variables, making it easier to solve for the other. This method involves modifying the equations, if necessary, so that when you add or subtract them, one of the variables is canceled out.

To utilize the addition method effectively:

• Ensure that the equations are in a standard form like Ax + By = C.
• Adjust the coefficients of one of the variables to become opposites. Sometimes you may need to multiply one or both equations by a suitable number to achieve this.
• Add the equations together to eliminate one of the variables, making it possible to solve for the remaining variable.

Once one variable is eliminated, you are left with a simple equation to solve. You can then back-substitute this solution into one of the original equations to find the value of the other variable.

Dependent Equations

Dependent equations occur when two equations in a linear system are linearly dependent, meaning one is a scalar multiple of the other. In simpler terms, dependent equations essentially represent the same line in a graph.

This happens when:

• The equations do not provide distinct information. They result in an identity – something always true, like 0 = 0.
• After eliminating variables using methods like addition, the remaining expression is true for all values of the variables that satisfy the equation.

When dealing with such equations, it's crucial to recognize that solving might not yield a single solution. Instead, the result will show that the system has infinitely many solutions, as both equations map to the same line.

Infinite Solutions

In a system of linear equations, infinite solutions occur when the equations represent the same line. Thus, every point on the line is a solution to both equations.

You know a system has infinite solutions when:

• Simplifying the equations using a method like addition results in a true identity, such as 0 = 0.
• Upon graphing, both equations overlap completely, forming a single line.

Infinite solutions indicate that the equations do not restrict the values to a single point pair solution. Instead, any point on the line described by these equations is a viable solution.

Linear Equations

Linear equations are equations of the first degree, meaning they involve constant terms and variables raised to the power of one. They create straight lines when graphed on a two-dimensional plane.

In a system of linear equations:

• The general form is Ax + By = C, where A, B, and C are constants.
• Each equation in the system shares variables that you need to solve for.
• The intersections of lines (if any) indicate potential solutions to the system.

By solving these systems, you can determine the specific values of the variables, which satisfy all equations simultaneously. Understanding linear equations is foundational for navigating through more complex algebraic concepts.