To solve the system of linear equations using the addition (elimination) method, follow these steps:

Given system:

1. $3x - y = 6$
2. $6x - 2y = 12$

Step 1: Rewrite the equations

• Equation 1: $3x - y = 6$
• Equation 2: $6x - 2y = 12$

Step 2: Align the coefficients for elimination

Multiply Equation 1 by 2 so that the $y$-terms have the same coefficient (in magnitude): $2 \cdot (3x - y) = 2 \cdot 6$ This gives: $6x - 2y = 12$

Now the system is:

1. $6x - 2y = 12$ (from scaling Equation 1)
2. $6x - 2y = 12$ (original Equation 2)

Step 3: Subtract one equation from the other

$(6x - 2y) - (6x - 2y) = 12 - 12$ $0 = 0$

Step 4: Interpret the result

The result $0 = 0$ indicates that the two equations are dependent, meaning they represent the same line. Therefore, there are infinitely many solutions, and the system is consistent with infinitely many solutions.

Solution:

The general solution can be expressed in parametric form: $x = t, \quad y = 3t - 6, \quad \text{where } t \text{ is any real number.}$

Would you like me to explain the parametric form further or provide graphical interpretation? 😊

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Related Questions:

1. How can we graphically interpret dependent systems in a coordinate plane?
2. What are the steps for solving linear systems with substitution instead of elimination?
3. How do we identify when a system has no solution during elimination?
4. What are the implications of infinite solutions in real-world applications of linear systems?
5. Can you explain how to write parametric solutions in matrix form?

Tip:

Always simplify equations as much as possible before applying elimination or substitution methods for efficiency and clarity.