Determine which pairs of equations have one solution no solution infinite solutions 2x+3=2x+7, 7x+3=7x+3, 4x+22=8x+10, 5x+5=5x+4, 3x+18=5x+8, 3(4x+8)=4(3x+6). Now what do you notice about equations that are in each of these categories?
- The pair of equations 2x+3=2x+7 and 7x+3=7x+3 both have no solution because in the first equation, the constant terms on both sides are not equal and in the second equation, the variables and constant terms are already equal on both sides.
- The pair of equations 4x+22=8x+10 and 5x+5=5x+4 both have one solution because in the first equation, when simplified, it becomes 4x - 8x = 10 - 22, which results in -4x = -12 and x = 3, and in the second equation, when simplified, it becomes 5x - 5x = 4 - 5, which results in 0 = -1, which is not true.
- The pair of equations 3x+18=5x+8 and 3(4x+8)=4(3x+6) both have infinite solutions because in the first equation, when simplified, it becomes 18 = 2x - x + 8, which results in 18 = x + 8, and in the second equation, when simplified, it becomes 12x + 24 = 12x + 24, which is already a true statement.
What is noticed about equations that are in each of these categories is:
- Equations with no solution have constant terms on both sides that are not equal.
- Equations with one solution have variables and constant terms on both sides that cancel out when simplified.
- Equations with infinite solutions result in a true statement when simplified.
Let's analyze each pair of equations and determine the number of solutions for each:
1. 2x + 3 = 2x + 7
This equation simplifies to 3 = 7, which is not true.
Conclusion: The pair of equations has no solution.
2. 7x + 3 = 7x + 3
This equation simplifies to 3 = 3, which is always true.
Conclusion: The pair of equations has infinitely many solutions.
3. 4x + 22 = 8x + 10
Subtracting 4x from both sides and simplifying gives 22 = 4x + 10.
Subtracting 10 from both sides gives 12 = 4x.
Dividing by 4 on both sides gives x = 3.
Conclusion: The pair of equations has one solution.
4. 5x + 5 = 5x + 4
This equation simplifies to 5 = 4, which is not true.
Conclusion: The pair of equations has no solution.
5. 3x + 18 = 5x + 8
Subtracting 3x from both sides and simplifying gives 18 = 2x + 8.
Subtracting 8 from both sides gives 10 = 2x.
Dividing by 2 on both sides gives x = 5.
Conclusion: The pair of equations has one solution.
6. 3(4x + 8) = 4(3x + 6)
Distributing on both sides gives 12x + 24 = 12x + 24.
Simplifying further, we can see that both sides are equal and true for all possible values of x.
Conclusion: The pair of equations has infinitely many solutions.
Based on the analysis of the given equations, we can notice the following about equations in each category:
- Equations with no solution have contradictory statements, such as 3 = 7 or 5 = 4.
- Equations with one solution have one unique value for the variable that satisfies the equation.
- Equations with infinitely many solutions have equivalent expressions on both sides of the equation, resulting in the equation always being true regardless of the value of x.
To determine whether each pair of equations has one solution, no solution, or infinite solutions, we can compare the coefficients of the variables.
1. 2x + 3 = 2x + 7:
- In this equation, the variables cancel out when we combine like terms.
- The equation simplifies to 3 = 7, which is false.
- Since the simplified equation is false, this pair of equations has no solution.
2. 7x + 3 = 7x + 3:
- In this equation, both sides are the same.
- Every value of x makes this equation true.
- Therefore, this pair of equations has infinite solutions.
3. 4x + 22 = 8x + 10:
- Here, we have variables and constants on both sides.
- By simplifying the equation, we get -6x = -12, which simplifies further to x = 2.
- As x has a specific value (x = 2) that satisfies the equation, this pair of equations has one solution.
4. 5x + 5 = 5x + 4:
- In this equation, the variables cancel out when we combine like terms.
- It simplifies to 5 = 4, which is false.
- Therefore, this pair of equations has no solution.
5. 3x + 18 = 5x + 8:
- Similar to the previous examples, by simplifying the equation, we get -2x = -10, which simplifies further to x = 5.
- Like earlier, x has a specific value (x = 5) that satisfies the equation, so this pair of equations has one solution.
6. 3(4x + 8) = 4(3x + 6):
- Distributing on both sides, we get 12x + 24 = 12x + 24.
- Here, the variable cancels out when we simplify further.
- Both sides of the equation are the same, indicating that every value of x makes this equation true.
- Thus, this pair of equations has infinite solutions.
From these examples, we notice that equations with no solution have simplified forms where a false statement is obtained (like 3 = 7). Equations with one solution have a specific value of x that satisfies the equation, and equations with infinite solutions have both sides of the equation be equivalent.