What does it mean to have no solutions?
Having no solutions means that there are no values or answers that satisfy a particular equation, problem, or condition. It indicates that there is not a valid solution within the given context or parameters. In mathematics, for example, if an equation has no solutions, it implies that the equation cannot be true for any value of the variable.
Consider the two equations that you underlined, which form a system with one
solution. Solve the equations. 3x+18=5x+8
3(4x8)=4(3x+)
To solve the system of equations:
Equation 1: 3x + 18 = 5x + 8
Equation 2: 3(4x + 8) = 4(3x + ?)
First, let's solve Equation 1:
Rearrange the equation by isolating the variable terms on one side:
3x - 5x = 8 - 18
-2x = -10
Divide both sides of the equation by -2 to solve for x:
x = -10 / -2
x = 5
Now that we have the value of x, we can substitute it into Equation 2 to find the value for the second variable:
3(4x + 8) = 4(3x + ?)
3(4(5) + 8) = 4(3(5) + ?)
3(20 + 8) = 4(15 + ?)
3(28) = 4(15 + ?)
84 = 60 + 4?
Subtract 60 from both sides of the equation:
84 - 60 = 4?
24 = 4?
Divide both sides by 4 to solve for ?:
? = 24 / 4
? = 6
Therefore, the solution for the system of equations is x = 5, ? = 6.
3(4x + 8) = 4(3x +6)
To solve the equation 3(4x + 8) = 4(3x + 6), we can follow these steps:
1. Distribute the coefficients:
12x + 24 = 12x + 24
2. Simplify the equation:
Both sides of the equation are identical, so they are equal.
In this case, the equation simplifies to 12x + 24 = 12x + 24.
When this occurs, it means that both sides of the equation are equivalent and will cancel out each other. Therefore, we do not have a unique solution in this case.
In conclusion, this equation has infinitely many solutions or no unique solution.
Having no solutions means that there is no possible value or set of values that can satisfy the given condition or equation. It implies that there is no solution or answer that can make the given condition or equation true. It is often indicated by terms such as "no solution," "inconsistent," or the graph of the equation not intersecting with the x-axis.
When we say that an equation or a problem has "no solutions," it means that there is no value or set of values that satisfies the conditions of the problem or equation. In other words, there is no possible answer that would make the equation or problem statement true.
To understand why a problem or equation has no solutions, we need to examine the conditions or constraints involved. It is often helpful to break down the problem step by step and analyze each component to determine where the contradiction or inconsistency arises.
For example, if we have a linear equation such as "2x + 3 = 2x + 5," we can subtract 2x from both sides to simplify the equation to "3 = 5." In this case, we can see that the equation leads to the contradiction that 3 is equal to 5, which is not possible. Therefore, the equation has no solutions.
In some cases, a problem may have no solutions due to conflicting conditions. For instance, if a problem states that "Find a positive number x such that x < 0," we immediately see that this is an impossible condition since a positive number cannot be less than zero. Hence, the problem has no solutions.
In summary, when we encounter a situation where there are no solutions, it means that the conditions or requirements specified in the problem or equation cannot be satisfied simultaneously.