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)
Let's analyze each pair of equations:
1) 2x+3=2x+7
This equation is inconsistent and has no solution. When we simplify it, we get:
2x+3 = 2x+7
-2x -2x
3 = 7
Since 3 is not equal to 7, there is no solution.
2) 7x+3=7x+3
This equation is always true and has infinite solutions. When we simplify it, we get:
7x+3 = 7x+3
-7x -7x
3 = 3
Since 3 is equal to 3, the equation is true for all values of x.
3) 4x+22=8x+10
This equation is consistent and has one solution. When we simplify it, we get:
4x+22 = 8x+10
-4x -4x
22 = 4x+10
-10 -10
12 = 4x
3 = x
So the solution is x = 3.
4) 5x+5=5x+4
This equation is inconsistent and has no solution. When we simplify it, we get:
5x+5 = 5x+4
-5x -5x
5 = 4
Since 5 is not equal to 4, there is no solution.
5) 3x+18=5x+8
This equation is consistent and has one solution. When we simplify it, we get:
3x+18 = 5x+8
-3x -3x
18 = 2x+8
-8 -8
10 = 2x
5 = x
So the solution is x = 5.
6) 3(4x+8)=4(3x+6)
This equation is consistent and has one solution. When we simplify it, we get:
12x+24 = 12x+24
The equation simplifies to 24 = 24, which is always true. This means that the equation is true for all values of x.
To summarize:
- Equations 2) and 6) have infinite solutions.
- Equations 1), 4), and 7) have no solution.
- Equations 3) and 5) have one solution.
Let's evaluate each pair of equations one by one:
1. 2x+3=2x+7
Rearrange the equation: 2x - 2x = 7 - 3
Simplify: 0 = 4
The variables x cancel out, and the equation simplifies to 0 = 4.
There is no solution to this equation since 0 is never equal to 4.
Therefore, this pair of equations has no solution.
2. 7x+3=7x+3
This equation is already simplified.
Both sides of the equation are the same.
Therefore, this pair of equations has infinite solutions.
3. 4x+22=8x+10
Rearrange the equation: 4x - 8x = 10 - 22
Simplify: -4x = -12
Divide both sides by -4: x = 3
The variable x has a specific value.
Therefore, this pair of equations has one solution.
4. 5x+5=5x+4
Rearrange the equation: 5x - 5x = 4 - 5
Simplify: 0 = -1
The variables x cancel out, and the equation simplifies to 0 = -1.
There is no solution to this equation since 0 is never equal to -1.
Therefore, this pair of equations has no solution.
5. 3x+18=5x+8
Rearrange the equation: 3x - 5x = 8 - 18
Simplify: -2x = -10
Divide both sides by -2: x = 5
The variable x has a specific value.
Therefore, this pair of equations has one solution.
6. 3(4x+8)=4(3x+6)
Distribute the multiplication: 12x + 24 = 12x + 24
Rearrange the equation: 12x - 12x = 24 - 24
Simplify: 0 = 0
The variables x cancel out, and the equation simplifies to 0 = 0.
Both sides of the equation are the same.
Therefore, this pair of equations has infinite solutions.
To summarize:
- Pair 1 has no solution.
- Pair 2 has infinite solutions.
- Pair 3 has one solution.
- Pair 4 has no solution.
- Pair 5 has one solution.
- Pair 6 has infinite solutions.
To determine the number of solutions for each pair of equations, we can simplify and compare the coefficients of the variables. Let's evaluate each pair one by one:
1. 2x+3=2x+7:
We can start by simplifying both sides of the equation:
2x - 2x + 3 = 2x - 2x + 7
3 = 7
Since we end up with a false statement (3 does not equal 7), this equation has no solution.
2. 7x+3=7x+3:
In this case, when we simplify both sides of the equation, we end up with the same expression on each side:
7x - 7x + 3 = 7x - 7x + 3
3 = 3
Since we end up with a true statement (3 equals 3), this equation has infinite solutions. Every value of x will satisfy the equation.
3. 4x+22=8x+10:
We can start by simplifying the equation as before:
4x - 8x = 10 - 22
-4x = -12
We have simplified the equation to -4x = -12. By dividing both sides by -4, we can determine the value of x:
-4x / -4 = -12 / -4
x = 3
Since we obtain a single value for x, this equation has one solution.
4. 5x+5=5x+4:
Similarly, we can simplify this equation:
5x - 5x = 4 - 5
0 = -1
Since we end up with a false statement (0 does not equal -1), this equation has no solution.
5. 3x+18=5x+8:
Let's simplify the equation:
3x - 5x = 8 - 18
-2x = -10
By dividing both sides by -2, we can find the value of x:
-2x / -2 = -10 / -2
x = 5
Therefore, this equation has one solution since x equals 5.
6. 3(4x+8)=4(3x+6):
Here we can distribute both sides of the equation to eliminate the parentheses:
12x + 24 = 12x + 24
After simplifying, we notice that both sides are identical. Whatever value we assign to x, the equation will always hold true.
Thus, this equation has infinite solutions.
To summarize:
- The equation 2x+3=2x+7 has no solution.
- The equation 7x+3=7x+3 has infinite solutions.
- The equation 4x+22=8x+10 has one solution, which is x = 3.
- The equation 5x+5=5x+4 has no solution.
- The equation 3x+18=5x+8 has one solution, which is x = 5.
- The equation 3(4x+8)=4(3x+6) has infinite solutions.