Select the correct choices to complete the sentence. Solve the equation and state whether the equation has one solution, no solution, or is an identity. -6y – 3 = 3 – 6y
The Correct Answer and Explanation is:
To solve the equation −6y−3=3−6y-6y – 3 = 3 – 6y−6y−3=3−6y, follow these steps:
Step 1: Simplify both sides
The equation is:−6y−3=3−6y-6y – 3 = 3 – 6y−6y−3=3−6y
Let’s start by adding 6y6y6y to both sides of the equation to eliminate the −6y-6y−6y term on the right-hand side:−6y+6y−3=3−6y+6y-6y + 6y – 3 = 3 – 6y + 6y−6y+6y−3=3−6y+6y
This simplifies to:−3=3-3 = 3−3=3
Step 2: Analyze the result
The equation now reads −3=3-3 = 3−3=3, which is clearly false. Since we arrive at a contradiction (a statement that is never true), this means there are no solutions to the equation.
Step 3: Conclusion
The equation −6y−3=3−6y-6y – 3 = 3 – 6y−6y−3=3−6y has no solution because we reached a contradiction. In general, if after simplifying an equation, we end up with an inconsistent statement like a=ba = ba=b where a≠ba \neq ba=b, then the equation has no solution.
If we had ended up with something like 0=00 = 00=0 after simplifying, it would have been an identity (meaning infinitely many solutions). However, since we reached a contradiction, the equation has no solution.
