Question

What do you notice about equations that are in each of these categories? Use complete sentences to answer the questions that follow.


Infinite solutions:
7x + 3 = 7x+3
3(4x + 8) = 4 ( 3x + 6)

No solution:
2x + 3 = 2x + 7
5x + 5 = 5x + 4

Answer 1

Equations that have infinite solutions typically express the same relationship in different forms, resulting in identities that are always true. For example, the equation \(7x + 3 = 7x + 3\) is true for any value of \(x\) because both sides are identical. Similarly, in the equation \(3(4x + 8) = 4(3x + 6)\), simplifying both sides leads to the same expression, confirming that there are infinitely many solutions.

On the other hand, equations that have no solution indicate a contradiction; they represent scenarios where two expressions cannot be equal regardless of the value of the variable. For instance, in the equation \(2x + 3 = 2x + 7\), when we attempt to solve for \(x\), we find that \(3\) cannot equal \(7\), indicating that there is no possible value of \(x\) that satisfies the equation. Similarly, in \(5x + 5 = 5x + 4\), subtracting \(5x\) from both sides leads to the statement \(5 = 4\), which is also a contradiction, confirming that this equation has no solutions.

In summary, infinite solutions arise from equivalent expressions, whereas no solutions are results of inherent contradictions within the equations.