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For each system of equations, drag the true statement about its solution set to the box under the system.
Infinitely Many Solutions: 6x + 3 is a multiple of 2x + 1.
Zero Solutions: y is equal to two different expressions.
Infinitely Many Solutions: Both equations are equal to y.
One Solution: 2x + 1 = 6x + 3 has one solution.
Zero Solutions: 6x – 3 can never be equal to 6x + 3.
Let’s analyze the statements given for the systems of equations:
1. Infinitely Many Solutions: 6x + 3 is a multiple of 2x + 1.
– This indicates that there are dependencies between the equations, suggesting that they overlap completely.
2. Zero Solutions: y is equal to two different expressions.
– If y must be equal to two different values at the same time, this creates a contradiction, leading to no solutions.
3. Infinitely Many Solutions: Both equations are equal to y.
– If both equations describe the same line, they will have infinitely many solutions.
4. One Solution: 2x + 1 = 6x + 3 has one solution.
– This means there is a unique point where these two lines intersect.
5. Zero Solutions: 6x – 3 can never be equal to 6x + 3.
– This indicates that the two sides of the equation never equal each other, confirming no solutions can exist.
Based on this analysis, you can match each statement to the appropriate box for the systems of equations:
– Infinitely Many Solutions: Both equations are equal to y.
– Zero Solutions: y is equal to two different expressions.
– One Solution: 2x + 1 = 6x + 3 has one solution.
– Zero Solutions: **6x –
To classify each system of equations based on the statements provided:
1. Infinitely Many Solutions: Both equations are equal to y.
If both equations are equal to y, it suggests they represent the same line, leading to infinitely many solutions.
2. Zero Solutions: 6x – 3 can never be equal to 6x + 3.
Simplifying this shows that the left side is always 6x – 3, while the right side is always 6x + 3. Since no x value can satisfy this equation, there are zero solutions.
3. One Solution: 2x + 1 = 6x + 3 has one solution.
This equation can be solved for x, resulting in a single solution.
4. Infinitely Many Solutions: 6x + 3 is a multiple of 2x + 1.
This suggests a relationship between the two equations that can lead to infinitely many solutions if they describe the same line in different forms.
5. Zero Solutions: y is equal to two different expressions.
If y is represented by two different expressions that do not intersect, this implies that the system has no points of intersection, hence no solution.
So, the correct arrangement would be:
– Infinitely Many Solutions: Both equations are equal to y.
– Zero Solutions: 6x – 3
For the system of equations provided, we can match the true statement to each case:
1. Infinitely Many Solutions: Both equations are equal to y.
– Explanation: If both equations represent the same line in a linear system, they will have infinitely many solutions because they overlap completely.
2. Zero Solutions: 6x – 3 can never be equal to 6x + 3.
– Explanation: This is true because simplifying both sides leads to a contradiction (subtracting 6x results in -3 = 3, which is false), indicating no solutions exist.
3. One Solution: 2x + 1 = 6x + 3 has one solution.
– Explanation: This equation can be solved algebraically, and it results in a single value for x, representing one unique solution.
4. Zero Solutions: y is equal to two different expressions.
– Explanation: If y is expressed as two different values (e.g., y = f(x) and y = g(x), where f(x) ≠ g(x)), it indicates that there is no point where both functions intersect, leading to zero solutions.
5. Infinitely Many Solutions: 6x + 3 is a multiple of 2x + 1.
– Explanation: This implies that one equation can be derived from the other through multiplication, suggesting infinitely many solutions as the lines coincide.
Overall, the correct matches
Let’s analyze the statements for each type of solution set in systems of equations:
1. Infinitely Many Solutions: “Both equations are equal to y.” – This means both equations are identical, leading to infinitely many solutions because every point on the line would satisfy both equations.
2. Zero Solutions: “6x – 3 can never be equal to 6x + 3.” – This shows a contradiction since there is no value of x that can satisfy this equation (the left side will always be 6x – 3, while the right will always be 6x + 3, leading to an impossible situation).
3. One Solution: “2x + 1 = 6x + 3 has one solution.” – This is true if you solve it. Rearranging gives you a single solution for x.
4. Zero Solutions: “y is equal to two different expressions.” – If y represents two different expressions, they can only intersect at most once, indicating inconsistency unless they are identical.
5. Infinitely Many Solutions: “6x + 3 is a multiple of 2x + 1.” – This statement is not necessarily always true; it depends on x, meaning there are cases that yield different values and not consistently multiple.
With that breakdown, you can match the statements accordingly. Let me know if you need further help with any specific part!