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+4, 3x+18=5x+8, 3(4x+8)=4(3x+6)
Let's solve each pair of equations to determine the number of solutions:
1) 2x + 3 = 2x + 7
Subtracting 2x from both sides:
3 = 7
This is a contradiction, so there are no solutions.
2) 7x + 3 = 7x + 3
The equations are identical. They have an infinite number of solutions.
3) 4x + 22 = 8x + 10
Subtracting 4x from both sides:
22 = 4x + 10
Subtracting 10 from both sides:
12 = 4x
Dividing both sides by 4:
3 = x
There is one solution.
4) 5x + 5 = 5x + 4
Subtracting 5x from both sides:
5 = 4
This is a contradiction, so there are no solutions.
5) 3x + 18 = 5x + 4
Subtracting 3x from both sides:
18 = 2x + 4
Subtracting 4 from both sides:
14 = 2x
Dividing both sides by 2:
7 = x
There is one solution.
6) 3x + 18 = 5x + 8
Subtracting 3x from both sides:
18 = 2x + 8
Subtracting 8 from both sides:
10 = 2x
Dividing both sides by 2:
5 = x
There is one solution.
7) 3(4x + 8) = 4(3x + 6)
Expanding the equation:
12x + 24 = 12x + 24
The equations are identical. They have an infinite number of solutions.
So, the pairs of equations that have one solution are 3) 4x + 22 = 8x + 10, 5) 3x + 18 = 5x + 4, and 6) 3x + 18 = 5x + 8.
The pairs of equations that have no solution are 1) 2x + 3 = 2x + 7 and 4) 5x + 5 = 5x + 4.
The pairs of equations that have an infinite number of solutions are 2) 7x + 3 = 7x + 3 and 7) 3(4x + 8) = 4(3x + 6).
Let's analyze each pair of equations one by one:
1. 2x+3=2x+7:
- Subtracting 2x from both sides: 3 = 7
- Since 3 is not equal to 7, this equation has no solution.
2. 7x+3=7x+3:
- Subtracting 7x from both sides: 3 = 3
- Since 3 is equal to 3, this equation has infinite solutions. The two sides of the equation are identical, so any value of x will satisfy it.
3. 4x+22=8x+10:
- Subtracting 4x from both sides and subtracting 10 from both sides: 12 = 4x
- Dividing both sides by 4: x = 3
- This equation has one solution, x = 3.
4. 5x+5=5x+4:
- Subtracting 5x from both sides: 5 = 4
- Since 5 is not equal to 4, this equation has no solution.
5. 3x+18=5x+4:
- Subtracting 3x from both sides and subtracting 4 from both sides: 14 = 2x
- Dividing both sides by 2: x = 7
- This equation has one solution, x = 7.
6. 3x+18=5x+8:
- Subtracting 3x from both sides and subtracting 8 from both sides: 10 = 2x
- Dividing both sides by 2: x = 5
- This equation has one solution, x = 5.
7. 3(4x+8)=4(3x+6):
- Distributing on both sides: 12x + 24 = 12x + 24
- The equation simplifies to 12x + 24 = 12x + 24, which means the two sides of the equation are identical.
- This equation has infinite solutions. Any value of x will satisfy it.
So, summarizing:
- The first pair has no solution.
- The second pair has infinite solutions.
- The third pair has one solution.
- The fourth pair has no solution.
- The fifth pair has one solution.
- The sixth pair has one solution.
- The seventh pair has infinite solutions.
To determine the number of solutions for each pair of equations, we need to simplify and analyze the equations individually.
1) 2x + 3 = 2x + 7
We have the same term (2x) on both sides of the equation. We can subtract 2x from both sides to eliminate it:
2x - 2x + 3 = 2x - 2x + 7
3 = 7
In this case, the variable x canceled out, leading to a contradiction (3 ≠ 7). Therefore, this pair of equations has no solution.
2) 7x + 3 = 7x + 3
In this case, the equation is 7x + 3 = 7x + 3, which means that both sides of the equation are identical. Since there are no variables to solve for, every value of x will satisfy this equation. Therefore, this pair of equations has infinite solutions.
3) 4x + 22 = 8x + 10
We can start by simplifying the equation by combining like terms:
4x - 8x = 10 - 22
-4x = -12
x = (-12) / (-4)
x = 3
In this case, x can be determined to be 3. Therefore, this pair of equations has one solution.
4) 5x + 5 = 5x + 4
We will start by simplifying the equation:
5x - 5x = 4 - 5
0 = -1
This equation leads to another contradiction (0 ≠ -1). Hence, this pair of equations has no solution.
5) 3x + 18 = 5x + 4
We'll simplify the equation by combining like terms:
3x - 5x = 4 - 18
-2x = -14
x = (-14) / (-2)
x = 7
In this case, the variable x can be calculated as 7. Therefore, this pair of equations has one solution.
6) 3x + 18 = 5x + 8
To simplify, we will combine like terms:
3x - 5x = 8 - 18
-2x = -10
x = (-10) / (-2)
x = 5
Here, x can be determined to be 5. So, this pair of equations has one solution.
7) 3(4x + 8) = 4(3x + 6)
We will start by distributing the multiplication:
12x + 24 = 12x + 24
This equation is similar to the second example. The equation is true for every value of x. Therefore, this pair of equations has infinite solutions.
To summarize the results:
- Pair 1: No solution
- Pair 2: Infinite solutions
- Pair 3: One solution
- Pair 4: No solution
- Pair 5: One solution
- Pair 6: One solution
- Pair 7: Infinite solutions