To determine whether Percy’s claim is correct, we need to analyze the system of equations provided:
- \( 6x + 4y = 14 \)
- \( 3x + 2y = k \)
First, we can manipulate the first equation to see if it can be expressed in terms of the second equation. We can simplify the first equation by dividing everything by 2:
\[ 3x + 2y = 7 \]
Now we have two equations:
- \( 3x + 2y = 7 \)
- \( 3x + 2y = k \)
From here, we can see that the left-hand sides of both equations are the same, while the right-hand sides are \( 7 \) and \( k \) respectively.
For the system of equations to have no solution, the two equations must represent parallel lines. This occurs when their slopes are equal, but their y-intercepts are different.
In terms of our equations:
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If \( k \neq 7 \), then the two equations represent parallel lines (since they have the same coefficients for \( x \) and \( y \) but different constant terms). Therefore, they would not intersect, leading to no solution.
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If \( k = 7 \), the two equations would be identical, resulting in infinitely many solutions (every point that satisfies one equation also satisfies the other).
Thus, Percy’s error lies in his assertion that any real number \( k \) would result in no solutions. The correct conclusion is that there would be no solutions only if \( k \neq 7\). If \( k = 7\), there are infinitely many solutions.
In summary, the claim that any real number \( k \) would cause the system to have no solution is incorrect because only specific values (any \( k \) not equal to 7) will produce that outcome.
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