Jillana begins to solve a linear equation that results in a variable expression set equal to the same variable expression. Which is the best interpretation of this solution?
The equation has one solution: [tex]$x=0$[/tex].
The equation has one solution: [tex]$x=1$[/tex].
The equation has no solution.
The equation has infinite solutions.
Answer (1)
When a linear equation results in a variable expression equal to itself, it simplifies to an identity.
An identity is true for all values of the variable.
Therefore, the equation has infinite solutions.
The best interpretation is that the equation has infinite solutions .
Explanation
Understanding the Result When Jillana solves a linear equation and arrives at a point where a variable expression is equal to the exact same variable expression, it means the equation simplifies to an identity. An identity is a statement that is always true, regardless of the value of the variable.
Example of an Identity For example, consider the equation 2 x + 3 = 2 x + 3 . No matter what value we substitute for x , the left side will always equal the right side. This is because the variable terms and the constant terms are identical on both sides of the equation.
Interpretation of Infinite Solutions When an equation simplifies to an identity, it means that any value of x will satisfy the equation. Therefore, the equation has infinitely many solutions.
Conclusion The best interpretation of this solution is that the equation has infinite solutions.
Examples
Imagine you're trying to divide a cake equally between two friends. If, no matter how you slice it, each friend always gets the same amount, then you have infinite ways to divide the cake fairly. Similarly, in math, if an equation is true for any value of the variable, it has infinite solutions.
