Question
2x+3=2x+7 5x+5=5x+4(no solution) 7x+3=7x+3 3(4x+8)=4(3x+6)(infinite solutions) 4x+22=8x+10 3x+18=5x+8(one solution) what do you notice about equations that are in each of these categories? Use complete sentences to answer the questions that follow.
Answers
GPT-4o mini
When examining the equations you've provided, we can categorize them based on the number of solutions they possess:
1. **No Solution**: The equations \( 2x + 3 = 2x + 7 \) and \( 5x + 5 = 5x + 4 \) both result in no solution. The first simplifies to \(3 = 7\) which is a false statement, indicating that no value of \(x\) will satisfy the equation. Similarly, the second equation simplifies to \(5 = 4\), also resulting in a contradiction. These types of equations typically occur when the variables on both sides are identical but the constants differ, making it impossible for the equation to hold true.
2. **Infinite Solutions**: The equation \( 3(4x + 8) = 4(3x + 6) \) is an example that has infinite solutions. When simplified, both sides yield the same expression, which implies that any value of \(x\) will satisfy the equation. Equations with infinite solutions essentially represent the same line when graphed, meaning that they are equivalent.
3. **One Solution**: The equations \( 4x + 22 = 8x + 10 \) and \( 3x + 18 = 5x + 8 \) are examples that have exactly one solution. These equations can be simplified and solved for \(x\) to yield a unique value. Typically, these equations represent distinct lines that intersect at a single point on a graph.
To summarize, equations can be categorized based on the number of solutions they offer: no solution indicates a contradiction, infinite solutions suggest that the equations are equivalent, and one solution indicates two distinct lines intersecting at a single point. This classification highlights the relationships between the coefficients and constants in linear equations.
1. **No Solution**: The equations \( 2x + 3 = 2x + 7 \) and \( 5x + 5 = 5x + 4 \) both result in no solution. The first simplifies to \(3 = 7\) which is a false statement, indicating that no value of \(x\) will satisfy the equation. Similarly, the second equation simplifies to \(5 = 4\), also resulting in a contradiction. These types of equations typically occur when the variables on both sides are identical but the constants differ, making it impossible for the equation to hold true.
2. **Infinite Solutions**: The equation \( 3(4x + 8) = 4(3x + 6) \) is an example that has infinite solutions. When simplified, both sides yield the same expression, which implies that any value of \(x\) will satisfy the equation. Equations with infinite solutions essentially represent the same line when graphed, meaning that they are equivalent.
3. **One Solution**: The equations \( 4x + 22 = 8x + 10 \) and \( 3x + 18 = 5x + 8 \) are examples that have exactly one solution. These equations can be simplified and solved for \(x\) to yield a unique value. Typically, these equations represent distinct lines that intersect at a single point on a graph.
To summarize, equations can be categorized based on the number of solutions they offer: no solution indicates a contradiction, infinite solutions suggest that the equations are equivalent, and one solution indicates two distinct lines intersecting at a single point. This classification highlights the relationships between the coefficients and constants in linear equations.