Question

1. santa claims that a linear equation can have exactly one solution, no solution, or infinitely many solutions. for each value of a, determine how many solutions would result for the linear equation 3x + 8 = a

exactly one solution	no solution	infinitely many solutions
a = 6x
a = 3x + 7
a = 3x + 8

Explanation:

Step 1: Analyze \(a = 0\)

The equation is \(3x + 8=0\). Solve for \(x\):
Subtract 8 from both sides: \(3x= - 8\)
Divide by 3: \(x =-\frac{8}{3}\). This is a unique solution, so exactly one solution.

Step 2: Analyze \(a = 6x\)

The equation is \(3x + 8=6x\)
Subtract \(3x\) from both sides: \(8 = 3x\)
Divide by 3: \(x=\frac{8}{3}\). This is a unique solution, so exactly one solution. Wait, there was a mistake earlier. Let's re - analyze:

Case 1: \(a = 0\)

Equation: \(3x+8 = 0\)
Subtract 8: \(3x=0 - 8=-8\)
Divide by 3: \(x =-\frac{8}{3}\). So exactly one solution.

Case 2: \(a = 6x\)

Equation: \(3x + 8=6x\)
Subtract \(3x\) from both sides: \(8=6x - 3x=3x\)
Divide by 3: \(x=\frac{8}{3}\). Exactly one solution.

Case 3: \(a = 3x + 7\)

Equation: \(3x+8=3x + 7\)
Subtract \(3x\) from both sides: \(8 = 7\). This is a false statement. So no solution.

Case 4: \(a=3x + 8\)

Equation: \(3x + 8=3x + 8\)
Subtract \(3x\) from both sides: \(8 = 8\). This is a true statement for all values of \(x\). So infinitely many solutions.

Answer:

• \(a = 0\): Exactly one solution.
• \(a = 6x\): Exactly one solution.
• \(a=3x + 7\): No solution.
• \(a = 3x+8\): Infinitely many solutions.

So the completed table is:

	Exactly one solution	No solution	Infinitely many solutions
\(a = 6x\)	\(\checkmark\)
\(a=3x + 7\)		\(\checkmark\)
\(a = 3x+8\)			\(\checkmark\)