investigate and explain the different types of solutions.
Question 1: Examine the six different equations that follow. Determine which pairs of equations have one solution, no solution, and 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)
To determine whether each pair of equations has one solution, no solution, or infinite solutions, we need to solve each pair of equations separately and analyze the solutions.
1) 2x + 3 = 2x + 7
Subtracting 2x from both sides, we get:
3 = 7
This equation simplifies to a contradiction, 3 does not equal 7. Therefore, this pair of equations has no solution.
2) 7x + 3 = 7x + 3
Subtracting 7x from both sides, we get:
3 = 3
This equation simplifies to an identity, 3 equals 3. Therefore, this pair of equations has infinite solutions.
3) 4x + 22 = 8x + 10
Subtracting 8x from both sides and subtracting 22 from both sides:
4x - 8x = 10 - 22
-4x = -12
Dividing both sides by -4:
x = 3
This equation has a single solution, x = 3. Therefore, this pair of equations has one solution.
4) 5x + 5 = 5x + 4
Subtracting 5x from both sides, we get:
5 = 4
This equation simplifies to a contradiction, 5 does not equal 4. Therefore, this pair of equations has no solution.
5) 3x + 18 = 5x + 8
Subtracting 3x from both sides, we get:
18 = 2x + 8
Subtracting 8 from both sides, we get:
10 = 2x
Dividing both sides by 2:
x = 5
This equation has a single solution, x = 5. Therefore, this pair of equations has one solution.
6) 3(4x + 8) = 4(3x + 6)
Expanding on both sides:
12x + 24 = 12x + 24
This equation simplifies to an identity, 12x + 24 equals 12x + 24. Therefore, this pair of equations has infinite solutions.
In summary:
- The pair of equations 2x + 3 = 2x + 7 has no solution.
- The pair of equations 7x + 3 = 7x + 3 has infinite solutions.
- The pair of equations 4x + 22 = 8x + 10 has one solution (x = 3).
- The pair of equations 5x + 5 = 5x + 4 has no solution.
- The pair of equations 3x + 18 = 5x + 8 has one solution (x = 5).
- The pair of equations 3(4x + 8) = 4(3x + 6) has infinite solutions.
what do you notice about equations that are in each of these categories? use complete sentences to answer the questions that follow.
From the given equations, we can observe the following patterns for equations in each category:
1) Equations with no solution:
In these equations, the expressions on both sides of the equation are equal initially, but they eventually lead to a contradiction (e.g., 3 = 7 or 5 = 4). This happens when the variables cancel out on both sides, resulting in an inconsistency. Therefore, no values of the variable will satisfy this type of equation.
2) Equations with infinite solutions:
In this case, the expressions on both sides of the equation are initially equal and remain equal throughout the entire equation-solving process (e.g., 7x + 3 = 7x + 3 or 3(4x + 8) = 4(3x + 6)). This implies that any value of the variable would make the equation true. Thus, infinitely many solutions exist for such equations.
3) Equations with one solution:
These equations include a specific value of the variable that satisfies both sides of the equation (e.g., 4x + 22 = 8x + 10 or 3x + 18 = 5x + 8). By performing mathematical operations, we obtain a unique solution for the variable, indicating that only one particular value can make the equation true.
consider the two equations: 7x + 3= 7x + 3 and 3(4x + 8)= 4(3x + 6). solve the equations
Let's solve the given equations:
1) 7x + 3 = 7x + 3
Subtracting 7x from both sides:
0 = 0
This equation simplifies to an identity, 0 equals 0. Since this equation is true for all values of x, it has infinite solutions.
2) 3(4x + 8) = 4(3x + 6)
Expanding on both sides:
12x + 24 = 12x + 24
We can see that the equation remains the same after expanding, which means that the equation is consistent and will be true for all values of x. Therefore, it has infinite solutions.
In summary, both equations 7x + 3 = 7x + 3 and 3(4x + 8) = 4(3x + 6) have infinite solutions.
what doses it mean to have infinite solutions?
Having infinite solutions means that the equation is true for any value of the variable. In other words, there are an infinite number of values that can be substituted into the equation, and the equation will still hold true. This typically occurs when both sides of the equation simplify to the same expression or when all terms cancel out, resulting in an identity.
In the context of the given equations, 7x + 3 = 7x + 3 and 3(4x + 8) = 4(3x + 6), having infinite solutions means that any value of x can be substituted into the equations, and they will remain true. This is because both equations simplify to an identity, indicating that they are true for all values of x.
consider the two equations: 3x + 3= 2x + 7 and 5x + 5= 5x + 4. solve the equations
Let's solve the given equations:
1) 3x + 3 = 2x + 7
Subtracting 2x from both sides and subtracting 3 from both sides:
3x - 2x = 7 - 3
x = 4
Therefore, the solution to the equation 3x + 3 = 2x + 7 is x = 4.
2) 5x + 5 = 5x + 4
Subtracting 5x from both sides and subtracting 5 from both sides:
5x - 5x = 4 - 5
0 = -1
Since 0 does not equal -1, this equation has no solutions.
In summary,
- The equation 3x + 3 = 2x + 7 has one solution: x = 4.
- The equation 5x + 5 = 5x + 4 has no solution.