Question

not all equations have exactly one solution. consider the equation 2n + 6 = 2(3 + n). can you find more solutions? complete the rest of the table.

n	solution?
1	?
2	?
3	?

Explanation:

Step1: Simplify the equation

First, simplify the right - hand side of the equation \(2n + 6=2(3 + n)\). Using the distributive property \(a(b + c)=ab+ac\), we have \(2(3 + n)=2\times3+2\times n = 6 + 2n\). So the equation becomes \(2n+6 = 2n + 6\).

Step2: Analyze the simplified equation

Subtract \(2n\) from both sides of the equation \(2n+6=2n + 6\). We get \((2n - 2n)+6=(2n - 2n)+6\), which simplifies to \(6 = 6\). This is a true statement for all values of \(n\).

Step3: Check for \(n = 1\)

Substitute \(n = 1\) into the original equation. Left - hand side: \(2\times1+6=2 + 6=8\). Right - hand side: \(2(3 + 1)=2\times4 = 8\). Since \(8 = 8\), \(n = 1\) is a solution.

Step4: Check for \(n = 2\)

Substitute \(n = 2\) into the original equation. Left - hand side: \(2\times2+6=4 + 6 = 10\). Right - hand side: \(2(3 + 2)=2\times5=10\). Since \(10 = 10\), \(n = 2\) is a solution.

Step5: Check for \(n = 3\)

Substitute \(n = 3\) into the original equation. Left - hand side: \(2\times3+6=6 + 6=12\). Right - hand side: \(2(3 + 3)=2\times6 = 12\). Since \(12 = 12\), \(n = 3\) is a solution.

Answer:

For \(n = 1\): solution; For \(n = 2\): solution; For \(n = 3\): solution