Question

no solution
one solution
infinitely many solutions
-1.7v + 2.8 = 1.4v - 3.1v + 2.8
4a - 3 + 2a = 7a - 2
\frac{1}{5}f - \frac{2}{3} = -\frac{1}{5}f + \frac{2}{3}
2y - 3 = 5 + 2(y - 1)
-3(n + 4) + n = -2(n + 6)

Explanation:

Step1: Analyze \(-1.7v + 2.8 = 1.4v - 3.1v + 2.8\)

Simplify right - hand side: \(1.4v-3.1v=-1.7v\), so the equation becomes \(-1.7v + 2.8=-1.7v + 2.8\). This is an identity (both sides are the same for all \(v\)), so it has infinitely many solutions.

Step2: Analyze \(4a - 3+2a = 7a - 2\)

Simplify left - hand side: \(4a + 2a-3=6a - 3\). The equation is \(6a-3 = 7a-2\). Subtract \(6a\) from both sides: \(-3=a - 2\). Add 2 to both sides: \(a=-1\). So this equation has one solution.

Step3: Analyze \(\frac{1}{5}f-\frac{2}{3}=-\frac{1}{5}f+\frac{2}{3}\)

Add \(\frac{1}{5}f\) to both sides: \(\frac{2}{5}f-\frac{2}{3}=\frac{2}{3}\). Add \(\frac{2}{3}\) to both sides: \(\frac{2}{5}f=\frac{4}{3}\). Multiply both sides by \(\frac{5}{2}\): \(f=\frac{4}{3}\times\frac{5}{2}=\frac{10}{3}\). So this equation has one solution.

Step4: Analyze \(2y - 3=5 + 2(y - 1)\)

Simplify right - hand side: \(5+2y-2=2y + 3\). The equation becomes \(2y-3 = 2y+3\). Subtract \(2y\) from both sides: \(-3 = 3\), which is a contradiction. So this equation has no solution.

Step5: Analyze \(-3(n + 4)+n=-2(n + 6)\)

Simplify left - hand side: \(-3n-12 + n=-2n-12\). Simplify right - hand side: \(-2n-12\). The equation is \(-2n-12=-2n-12\), which is an identity (true for all \(n\)), so it has infinitely many solutions.

Answer:

• No Solution: \(2y - 3 = 5+2(y - 1)\)
• One Solution: \(4a - 3 + 2a=7a - 2\), \(\frac{1}{5}f-\frac{2}{3}=-\frac{1}{5}f+\frac{2}{3}\)
• Infinitely Many Solutions: \(-1.7v + 2.8 = 1.4v-3.1v + 2.8\), \(-3(n + 4)+n=-2(n + 6)\)