Question

1. select whether each equation has no solution, one solution, or infinitely many solutions

2.5(n + 4)=n + 1.5n - 7
2.5(n + 4)=2n + 0.5n + 10
2.5(n + 4)=2.5n + 0.5n
determine the measure in degrees of the angle indicated with an arc.
11.
12.

Explanation:

Step1: Solve the first equation \(2.5(n + 4)=n + 1.5n-7\)

Expand the left - hand side: \(2.5n+10 = 2.5n-7\).
Subtract \(2.5n\) from both sides: \(10=-7\). Since this is a contradiction, there is no solution.

Step2: Solve the second equation \(2.5(n + 4)=2n + 0.5n+10\)

Expand the left - hand side: \(2.5n + 10=2.5n+10\).
Subtract \(2.5n\) from both sides: \(10 = 10\). This is an identity, so there are infinitely many solutions.

Step3: Solve the third equation \(2.5(n + 4)=2.5n+0.5n\)

Expand the left - hand side: \(2.5n+10=3n\).
Subtract \(2.5n\) from both sides: \(10 = 0.5n\).
Divide both sides by \(0.5\): \(n = 20\). So there is one solution.

Step4: Solve problem 11

We know that \(2x+82.5 + 3x=180\) (since the sum of angles on a straight line is \(180^{\circ}\)).
Combine like terms: \(5x+82.5 = 180\).
Subtract \(82.5\) from both sides: \(5x=180 - 82.5=97.5\).
Divide both sides by \(5\): \(x = 19.5\).
The angle we want is \(3x\), so \(3\times19.5 = 58.5^{\circ}\).

Step5: Solve problem 12

We know that \(7x+8x+90 = 180\) (since the sum of angles around a right - angle intersection of two lines: the sum of angles in the non - right - angle part is \(180^{\circ}\)).
Combine like terms: \(15x+90 = 180\).
Subtract \(90\) from both sides: \(15x=90\).
Divide both sides by \(15\): \(x = 6\).
The angle we want is \(7x\), so \(7\times6=42^{\circ}\).

Answer:

• For \(2.5(n + 4)=n + 1.5n-7\): No solution.
• For \(2.5(n + 4)=2n + 0.5n+10\): Infinitely many solutions.
• For \(2.5(n + 4)=2.5n+0.5n\): One solution.
• Problem 11: \(58.5^{\circ}\).
• Problem 12: \(42^{\circ}\).