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For each system of equations, drag the true statement about its solution set to the box under the system.
Zero Solutions:
4x + 2 can never be equal to 4x – 2.
Zero Solutions:
y is equal to two different expressions.
One Solution:
3x – 4 = 2x + 2 has one solution.
Infinitely Many Solutions:
4x + 2 is a multiple of 2x + 1.
Infinitely Many Solutions:
Both equations are equal to y.
To analyze the provided system of equations, let’s break down each statement:
1. Zero Solutions:
– “4x + 2 can never be equal to 4x – 2.”
– This statement is true because when you set (4x + 2 = 4x – 2) and simplify, you get (2 = -2), which is a contradiction.
2. Zero Solutions:
– “y is equal to two different expressions.”
– If we have the same variable (y) defined by two different expressions that are not equal, it also results in a contradiction.
3. One Solution:
– “3x – 4 = 2x + 2 has one solution.”
– This can be solved: (3x – 2x = 4 + 2) simplifies to (x = 6), thereby confirming there is one unique solution.
4. Infinitely Many Solutions:
– “4x + 2 is a multiple of 2x + 1.”
– If one expression can be expressed as a multiple of another, it often implies they share the same linear relationship, leading to infinitely many solutions.
5. Infinitely Many Solutions:
– “Both equations are equal to y.”
– If two different equations are set equal to (y) and are identical, they correspond to the same graph and provide infinitely
Let’s analyze each of the statements related to the systems of equations:
1. Zero Solutions:
– “4x + 2 can never be equal to 4x – 2.”
– This is true because if we subtract 4x from both sides, we get 2 = -2, which is a false statement. Hence, there are no solutions.
2. Zero Solutions:
– “y is equal to two different expressions.”
– This is true because if y is defined by two different equations that don’t intersect, they cannot be equal, leading to zero solutions.
3. One Solution:
– “3x – 4 = 2x + 2 has one solution.”
– This is true. If we solve for x, we subtract 2x from both sides to get x – 4 = 2, then by adding 4, we find x = 6. There’s only one solution.
4. Infinitely Many Solutions:
– “4x + 2 is a multiple of 2x + 1.”
– This requires more context. If it means the equations are dependent, there could be infinitely many solutions if one equation can be derived from the other.
5. Infinitely Many Solutions:
– “Both equations are equal to y.”
– This statement implies that both equations represent the same line when graphed, indicating there are infinitely many points (solutions
To determine the correct statements about the solution sets for each system of equations, we can analyze each statement:
1. Zero Solutions:
– “4x + 2 can never be equal to 4x – 2.”
– This is true, as simplifying shows that it leads to a contradiction (2 ≠ -2).
2. Zero Solutions:
– “y is equal to two different expressions.”
– This generally means that the two expressions cannot equal each other, resulting in no solution.
3. One Solution:
– “3x – 4 = 2x + 2 has one solution.”
– This can be solved to find that x = 6 is a valid solution, making this statement true.
4. Infinitely Many Solutions:
– “4x + 2 is a multiple of 2x + 1.”
– This suggests a proportional relationship; if true, this can yield infinitely many solutions.
5. Infinitely Many Solutions:
– “Both equations are equal to y.”
– If two equations are both equal to y, they represent the same line, thus having infinitely many solutions wherever they overlap.
### Summary:
– Zero Solutions: “4x + 2 can never be equal to 4x – 2.” & “y is equal to two different expressions.”
– One Solution
To address your question about the systems of equations, let’s analyze each statement and match them with the correct solution set.
1. Zero Solutions:
– 4x + 2 can never be equal to 4x – 2.
– This is true because when you simplify this equation, you would get 2 = -2, which is a contradiction, indicating that there are no solutions.
2. Zero Solutions:
– y is equal to two different expressions.
– This typically indicates that the two expressions represent lines that are parallel and never intersect, leading to zero solutions.
3. One Solution:
– 3x – 4 = 2x + 2 has one solution.
– You can solve this equation: 3x – 2x = 2 + 4, which simplifies to x = 6. Thus, it has one unique solution.
4. Infinitely Many Solutions:
– 4x + 2 is a multiple of 2x + 1.
– If you express 4x + 2 as a multiple of 2x + 1, you can show that they’re essentially equivalent in some multiples, leading to infinitely many solutions since the two lines would overlap.
5. Infinitely Many Solutions:
– Both equations are equal to y.
– If two equations are both expressed as