Match the description of the one variable equation with the number of solutions it will have.(4 points)
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x−7=7−x
3(x+5)=3x+5
10−x=25
2(x+4)=2x+5+3
x−7=7−x - This equation will have one solution.
3(x+5)=3x+5 - This equation will have infinitely many solutions.
10−x=25 - This equation will have no solution.
2(x+4)=2x+5+3 - This equation will have one solution.
x-7=7-x -> Solution: 1 solution.
3(x+5)=3x+5 -> Solution: Infinitely many solutions.
10-x=25 -> Solution: No solution.
2(x+4)=2x+5+3 -> Solution: 1 solution.
x−7=7−x: This equation has one solution.
3(x+5)=3x+5: This equation has infinitely many solutions.
10−x=25: This equation has no solution.
2(x+4)=2x+5+3: This equation has no solution.
To determine the number of solutions for each equation, we need to solve them and analyze the results.
1. x - 7 = 7 - x
To solve this equation, we can simplify it by combining like terms:
2x = 14
Dividing both sides by 2:
x = 7
This equation has a unique solution since there is only one value for x that satisfies it.
Therefore, the first equation has 1 solution.
2. 3(x + 5) = 3x + 5
Distributing 3 to both terms inside the parentheses:
3x + 15 = 3x + 5
Subtracting 3x from both sides:
15 = 5
This equation leads to a contradiction, as there is no value for x that satisfies it. The equation is inconsistent.
As a result, the second equation has 0 solutions.
3. 10 - x = 25
To solve this equation, we can isolate the variable x by moving constants to the other side:
-x = 25 - 10
-x = 15
Multiplying both sides by -1 to switch the sign:
x = -15
This equation has a unique solution since there is only one value for x that satisfies it.
Hence, the third equation has 1 solution.
4. 2(x + 4) = 2x + 5 + 3
Expanding and simplifying:
2x + 8 = 2x + 8
Subtracting 2x from both sides:
8 = 8
This equation is true for any value of x since the variable cancels out.
Therefore, the fourth equation has infinitely many solutions.
To summarize:
- Equation 1: 1 solution
- Equation 2: 0 solutions
- Equation 3: 1 solution
- Equation 4: infinitely many solutions