2x + 3 = 2x + 7 7x + 3 = 7x + 3 4x + 22 = 8x + 10
5x + 5 = 5x + 4 3x + 18 = 5x + 8 3(4x + 8) = 4(3x + 6)
What do you notice about equations that are in each of these categories? Use complete sentences to answer the questions that follow.
In each of these categories of equations, there is a constant term on both sides of the equation. This means that the equation is balanced and there is an equal amount of terms on both sides. Additionally, in each equation, the variable term cancels out when simplifying both sides of the equation. This implies that the variable does not play a role in determining the solutions to the equation.
which of the two equasions have no solution
The equations that have no solution are:
7x + 3 = 7x + 3
3(4x + 8) = 4(3x + 6)
In both of these equations, the variable term cancels out when simplifying both sides, resulting in a constant term on both sides of the equation. This means that the equation is balanced, but there are no specific values for the variable that will make the equation true. Since there are no values that satisfy the equation, these equations have no solution.
which two have infinant solutions
The equations that have infinitely many solutions are:
2x + 3 = 2x + 7
5x + 5 = 5x + 4
In both of these equations, when we simplify both sides of the equation, we get a true statement such as 0 = 0. This means that any value of x will make the equation true. Thus, these equations have infinitely many solutions.