Questions: 2x+6y=8 3x+9y=12

Transcript text: \[ \begin{array}{c} 2 x+6 y=8 \\ 3 x+9 y=12 \end{array} \]

Solution

Solution Steps

To solve the given system of linear equations, we can use the method of elimination or substitution. Here, we will use the elimination method. First, we will multiply the first equation by 3 and the second equation by 2 to make the coefficients of \(x\) the same. Then, we will subtract one equation from the other to eliminate \(x\) and solve for \(y\). Once \(y\) is found, we will substitute it back into one of the original equations to find \(x\).

Step 1: Analyze the System of Equations

The given system of equations is: \[ \begin{align_} 2x + 6y &= 8 \\ 3x + 9y &= 12 \end{align_} \]

Step 2: Simplify the Equations

Both equations can be simplified by dividing through by their greatest common divisors:

The first equation can be divided by 2: \[ x + 3y = 4 \]
The second equation can be divided by 3: \[ x + 3y = 4 \]

Step 3: Identify the Relationship

Both simplified equations are identical, indicating that they represent the same line. Therefore, there are infinitely many solutions along the line \(x + 3y = 4\).

Step 4: Express \(x\) in Terms of \(y\)

From the equation \(x + 3y = 4\), we can express \(x\) in terms of \(y\): \[ x = 4 - 3y \]

Final Answer

The solution to the system of equations is a line described by: \[ \boxed{x = 4 - 3y} \] This indicates that for any value of \(y\), \(x\) can be determined using the equation \(x = 4 - 3y\).

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