Question
Which of the following statements are true? Select all the correct answers. If you rewrite a one-variable linear equation and see a statement like \( 4=4 \) or \( 4 a+6=4 a+6 \), then the equation has infinitely many solutions. B If you rewrite a one-variable linear equation and the variable terms are the same on each side of the equation, then you can solve the equation and find the value of the variable. C If a one-variable linear equation has one solution, then every value of the variable makes the equation true. D If both sides of a one-variable linear equation have the same variable term and different constant terms, then the equation has infinitely many solutions. E If a one-variable linear equation has no solution, then no value of the variable will make the equation true.
Ask by Potter Fuentes. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The correct answers are A and E.
Solution
Let's analyze each statement one by one to determine their truthfulness.
**Statement A:** If you rewrite a one-variable linear equation and see a statement like \( 4=4 \) or \( 4a+6=4a+6 \), then the equation has infinitely many solutions.
- This statement is **true**. When both sides of the equation are identical, it means that the equation holds for all values of the variable, leading to infinitely many solutions.
**Statement B:** If you rewrite a one-variable linear equation and the variable terms are the same on each side of the equation, then you can solve the equation and find the value of the variable.
- This statement is **false**. If the variable terms are the same on both sides, the equation may reduce to a statement about constants. For example, if you have \( 3x + 2 = 3x + 5 \), it simplifies to \( 2 = 5 \), which is false. Thus, you cannot always find a unique solution.
**Statement C:** If a one-variable linear equation has one solution, then every value of the variable makes the equation true.
- This statement is **false**. If an equation has one solution, it means that only one specific value of the variable satisfies the equation, not every value.
**Statement D:** If both sides of a one-variable linear equation have the same variable term and different constant terms, then the equation has infinitely many solutions.
- This statement is **false**. If the variable terms are the same but the constant terms are different, the equation will lead to a contradiction. For example, \( 2x + 3 = 2x + 5 \) simplifies to \( 3 = 5 \), which is false. Therefore, there are no solutions.
**Statement E:** If a one-variable linear equation has no solution, then no value of the variable will make the equation true.
- This statement is **true**. If an equation has no solution, it means that there is no value of the variable that can satisfy the equation.
### Summary of Truthfulness:
- **A:** True
- **B:** False
- **C:** False
- **D:** False
- **E:** True
Thus, the correct answers are **A** and **E**.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
When you simplify a one-variable linear equation and end up with an identity like \( 4=4 \) or \( 4a+6=4a+6 \), it’s like finding a treasure chest with an infinite stash of gold—meaning the equation has infinitely many solutions! These scenarios indicate that every number you plug in will fulfill the equation. On the flip side, if you get an equation such as \( 4a + 6 = 4a + 7 \), where the variable terms are identical but the constants differ, it's like trying to fit a square peg into a round hole—this equation has no solution. It creates a contradiction, which is impossible to satisfy, so no value can ever make the equation true!
