Question

1. which one is infinite, no solution and 1 solution (matching) *
2. dependent
3. $5x + 10 = 6x - 20$
4. inconsistent
5. $6(x + 3) = 6x + 18$
6. independent
7. select answer

$5x + 10 = 6x - 20$
$9x + 7 = 9x$
$6(x + 3) = 6x + 18$

Explanation:

To solve this matching problem, we analyze each equation and pair it with the appropriate term (Dependent, Inconsistent, Independent) based on the number of solutions:

Step 1: Analyze \( 5x + 10 = 6x - 20 \)

Subtract \( 5x \) from both sides: \( 10 = x - 20 \). Then add 20: \( x = 30 \).
This equation has 1 solution (a unique solution), so it is Independent (Independent systems have one solution).

Step 2: Analyze \( 6(x + 3) = 6x + 18 \)

Expand the left side: \( 6x + 18 = 6x + 18 \).
Subtract \( 6x \) from both sides: \( 18 = 18 \) (always true).
This equation has infinite solutions (the equations are equivalent), so it is Dependent (Dependent systems have infinite solutions).

Step 3: Analyze \( 9x + 7 = 9x \) (from the dropdown)

Subtract \( 9x \) from both sides: \( 7 = 0 \) (never true).
This equation has no solution (the equations are contradictory), so it is Inconsistent (Inconsistent systems have no solution).

Final Matching:

1. Dependent → \( 6(x + 3) = 6x + 18 \)
2. Inconsistent → \( 9x + 7 = 9x \)
3. Independent → \( 5x + 10 = 6x - 20 \)

Answer:

1. Dependent - \( 6(x + 3) = 6x + 18 \)
2. Inconsistent - \( 9x + 7 = 9x \)
3. Independent - \( 5x + 10 = 6x - 20 \)