Determine which pairs of equations have one solution, no solution, and infinite solutions. Two will have one solution, two will have no solution, and two will have an infinite solution. 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 solve each pair of equations:
1) 2x+3=2x+7
Subtract 2x from both sides:
3 = 7
Since 3 does not equal 7, this pair of equations has no solution.
2) 7x+3=7x+3
Subtract 7x from both sides:
3 = 3
Since 3 is equal to 3, this pair of equations has an infinite number of solutions.
3) 4x+22=8x+10
Subtract 4x from both sides:
22 = 4x+10
Subtract 10 from both sides:
12 = 4x
Divide both sides by 4:
3 = x
Since x has a specific value (x = 3), this pair of equations has one solution.
4) 5x+5=5x+4
Subtract 5x from both sides:
5 = 4
Since 5 does not equal 4, this pair of equations has no solution.
5) 3x+18=5x+8
Subtract 3x from both sides:
18 = 2x+8
Subtract 8 from both sides:
10 = 2x
Divide both sides by 2:
5 = x
Since x has a specific value (x = 5), this pair of equations has one solution.
6) 3(4x+8)=4(3x+6)
Distribute the 3 on the left side:
12x+24 = 4(3x+6)
Distribute the 4 on the right side:
12x+24 = 12x+24
Subtract 12x from both sides:
24 = 24
Since 24 is equal to 24, this pair of equations has an infinite number of solutions.
So, the pairs of equations with one solution are:
- 4x+22=8x+10
- 5x+5=5x+4
The pairs with no solution are:
- 2x+3=2x+7
- 5x+5=5x+4
The pairs with an infinite number of solutions are:
- 7x+3=7x+3
- 3(4x+8)=4(3x+6)
Let's analyze each pair of equations one by one:
1. 2x + 3 = 2x + 7:
If we simplify this equation, we get 2x - 2x = 7 - 3, which simplifies to 0 = 4. This is not possible, as zero cannot be equal to four. Therefore, this pair of equations has no solution.
2. 7x + 3 = 7x + 3:
Both sides of the equation are the same, meaning they are equal for all values of x. This pair of equations has infinite solutions.
3. 4x + 22 = 8x + 10:
Let's solve this equation step-by-step:
Moving the variables to one side, we get 4x - 8x = 10 - 22.
Simplifying further, we have -4x = -12.
Dividing both sides by -4, we get x = 3.
This pair of equations has one solution.
4. 5x + 5 = 5x + 4:
If we simplify this equation, we get 5x - 5x = 4 - 5, which simplifies to 0 = -1. This is not possible, as zero cannot be equal to negative one. Therefore, this pair of equations has no solution.
5. 3x + 18 = 5x + 8:
Let's solve this equation step-by-step:
Moving the variables to one side, we get 3x - 5x = 8 - 18.
Simplifying further, we have -2x = -10.
Dividing both sides by -2, we get x = 5.
This pair of equations has one solution.
6. 3(4x + 8) = 4(3x + 6):
Let's distribute the terms and solve this equation step-by-step:
12x + 24 = 12x + 24.
Both sides of the equation are the same, meaning they are equal for all values of x. This pair of equations has infinite solutions.
To summarize:
- The equations 2x + 3 = 2x + 7 and 4x + 22 = 8x + 10 have no solutions.
- The equations 7x + 3 = 7x + 3 and 3(4x + 8) = 4(3x + 6) have infinite solutions.
- The equations 5x + 5 = 5x + 4 and 3x + 18 = 5x + 8 have one solution each.
To determine whether a pair of equations has one solution, no solution, or infinite solutions, we can start by simplifying the equations and comparing them.
1. 2x+3=2x+7
Simplifying, we notice that both sides of the equation have identical expressions (2x). Therefore, this equation has infinite solutions.
2. 7x+3=7x+3
Simplifying, we see that both sides of the equation are equal. In other words, every possible value of x will make the equation true. This equation also has infinite solutions.
3. 4x+22=8x+10
By simplifying and rearranging, we can find the solution:
4x - 8x = 10 - 22
-4x = -12
Dividing both sides by -4, we find:
x = 3
Since there is a specific value of x that satisfies the equation, this pair of equations has one solution.
4. 5x+5=5x+4
Simplifying, we can see that the variables will cancel out:
5x - 5x = 4 - 5
0 = -1
This is a contradiction (0 cannot be equal to -1). Therefore, this pair of equations has no solution.
5. 3x+18=5x+8
By simplifying and rearranging, we can find the solution:
3x - 5x = 8 - 18
-2x = -10
Dividing both sides by -2, we find:
x = 5
As there is a specific value of x that satisfies the equation, this pair of equations has one solution.
6. 3(4x+8)=4(3x+6)
Expanding and simplifying, we can solve for x:
12x + 24 = 12x + 24
The equation on both sides is identical, so every value of x will make this equation true. Therefore, this equation has infinite solutions.
In summary:
- Equations 1 and 2 have infinite solutions.
- Equations 3 and 5 have one solution.
- Equations 4 and 6 have no solution.