Question

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

$9x + 7 = 9x$

1. $6(x + 3) = 6x + 18$

Explanation:

for Each Equation:

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

Step 1: Solve for \(x\)

Subtract \(5x\) from both sides: \(10 = x - 20\)

Step 2: Isolate \(x\)

Add 20 to both sides: \(x = 30\)
This equation has one solution, so it matches with "Independent" (since independent systems of equations (or single equations here in the context of linear equations) can have one solution).

Equation 2: \(9x + 7 = 9x\)

Step 1: Subtract \(9x\) from both sides

\(7 = 0\), which is a false statement.
This equation has no solution, so it matches with "Inconsistent" (inconsistent systems have no solutions).

Equation 3: \(6(x + 3) = 6x + 18\)

Step 1: Expand the left - hand side

\(6x+18 = 6x + 18\)

Step 2: Subtract \(6x\) and 18 from both sides (or observe)

We get \(0 = 0\), which is always true. This equation has infinite solutions, so it matches with "Dependent" (dependent systems have infinite solutions).

Answer:

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