To identify pairs of linear equations that have the same solution set, we must consider equations that are essentially equivalent. This means that one equation can be derived from the other by multiplying or dividing by a non-zero constant, adding or subtracting the same term from both sides, or by applying any other algebraic manipulation that doesn't change the solution.
For example, consider the following two equations:
- \( 2x + 4y = 8 \)
- \( x + 2y = 4 \)
To see if these equations have the same solution set, let's manipulate the first equation:
- Divide the entire equation \( 2x + 4y = 8 \) by 2:
\[ x + 2y = 4 \]
Now we have transformed the first equation into the second, indicating that both equations indeed describe the same line in the coordinate plane. Therefore, they share the same solution set.
Another example could be:
- \( 3x - 6y = 12 \)
- \( x - 2y = 4 \)
To check for equivalence, let's transform the first equation by dividing by 3:
- Dividing \( 3x - 6y = 12 \) by 3 gives:
\[ x - 2y = 4 \]
Again, we derive the second equation from the first, confirming they have the same solution set.
Overall, pairs of linear equations that are scalar multiples or can be manipulated into each other will have the same solution set. You can work through similar equations by considering constants and arithmetic operations to verify equivalency.
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