Question

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

\t\t\t\t\t\t\t\t $9x + 7 = 9x$
\t\t\t\t\t\t\t\t $6(x + 3) = 6x + 18$

Explanation:

To solve this matching problem, we analyze each type of system (dependent, inconsistent, independent) and the equations:

1. Dependent (Infinite Solutions)

A dependent system has equations that are equivalent (same line), so they have infinite solutions. Let's check the equations:

• \( 6(x + 3) = 6x + 18 \)

Simplify left side: \( 6x + 18 = 6x + 18 \). Both sides are identical, so this equation is always true (infinite solutions). Thus, Dependent matches \( 6(x + 3) = 6x + 18 \).

2. Inconsistent (No Solution)

An inconsistent system has equations that are contradictory (parallel lines, no intersection). Let's check:

• \( 9x + 7 = 9x \)

Subtract \( 9x \) from both sides: \( 7 = 0 \), which is false. Thus, this equation has no solution. So, Inconsistent matches \( 9x + 7 = 9x \).

3. Independent (One Solution)

An independent system has one unique solution (intersecting lines). Let's check:

• \( 5x + 10 = 6x - 20 \)

Subtract \( 5x \) from both sides: \( 10 = x - 20 \). Add 20 to both sides: \( x = 30 \). This has one unique solution. Thus, Independent matches \( 5x + 10 = 6x - 20 \).

Final Matches:

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

(For the dropdown in the image:

• The second dropdown (labeled "2.") should be \( 9x + 7 = 9x \) (Inconsistent).
• The third equation (labeled "3.") is \( 6(x + 3) = 6x + 18 \) (Dependent).)

Answer:

To solve this matching problem, we analyze each type of system (dependent, inconsistent, independent) and the equations:

1. Dependent (Infinite Solutions)

A dependent system has equations that are equivalent (same line), so they have infinite solutions. Let's check the equations:

• \( 6(x + 3) = 6x + 18 \)

Simplify left side: \( 6x + 18 = 6x + 18 \). Both sides are identical, so this equation is always true (infinite solutions). Thus, Dependent matches \( 6(x + 3) = 6x + 18 \).

2. Inconsistent (No Solution)

An inconsistent system has equations that are contradictory (parallel lines, no intersection). Let's check:

• \( 9x + 7 = 9x \)

Subtract \( 9x \) from both sides: \( 7 = 0 \), which is false. Thus, this equation has no solution. So, Inconsistent matches \( 9x + 7 = 9x \).

3. Independent (One Solution)

An independent system has one unique solution (intersecting lines). Let's check:

• \( 5x + 10 = 6x - 20 \)

Subtract \( 5x \) from both sides: \( 10 = x - 20 \). Add 20 to both sides: \( x = 30 \). This has one unique solution. Thus, Independent matches \( 5x + 10 = 6x - 20 \).

Final Matches:

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

(For the dropdown in the image:

• The second dropdown (labeled "2.") should be \( 9x + 7 = 9x \) (Inconsistent).
• The third equation (labeled "3.") is \( 6(x + 3) = 6x + 18 \) (Dependent).)