Which of the following is an example of an equation with no solution?(1 point)Responses3x+4=3x+43 x plus 4 equals 3 x plus 43x+4=4x+33 x plus 4 equals 4 x plus 34x+3=3x+34 x plus 3 equals 3 x plus 33x+4=3x+33 x plus 4 equals 3 x plus 3
3x+4=3x+4 is an example of an equation with no solution.
Which of the following is an example of an equation with exactly one solution?(1 point)Responses3x+4=3x+43 x plus 4 equals 3 x plus 44x+3=4x+34 x plus 3 equals 4 x plus 33x+4=3x+33 x plus 4 equals 3 x plus 33x+4=4x+33 x plus 4 equals 4 x plus 3
3x+4=4x+3 is an example of an equation with exactly one solution.
Solve for x to determine if the equation has one, none, or infinite solutions. 3(x+2)−6=3x (1 point)ResponsesThe equation has no solution.The equation has no solution.The equation has one solution: x=3.The equation has one solution: x equals 3 .The equation has one solution: x=−3.The equation has one solution: x equals negative 3 .The equation has infinite solutions.
Let's solve the equation step by step:
3(x+2) - 6 = 3x
Distribute the 3 on the left side:
3x + 6 - 6 = 3x
Combine like terms:
3x = 3x
Subtract 3x from both sides:
0 = 0
Since 0 is equal to 0, this means that the equation has infinite solutions.
Solve for x to determine if the equation has one, none, or infinite solutions. 4x=3(x+5)−15 (1 point)ResponsesThe equation has infinite solutions.The equation has infinite solutions.The equation has no solution.The equation has no solution.The equation has one solution: x=0.The equation has one solution: x equals 0 .The equation has one solution: x=−10.
Let's solve the equation step by step:
4x = 3(x+5) - 15
Distribute the 3 on the right side:
4x = 3x + 15 - 15
Simplify:
4x = 3x
Subtract 3x from both sides:
x = 0
Since x simplifies to a single value (x = 0), this means that the equation has one solution.
Solve for x to determine if the equation has one, none, or infinite solutions. 5(2x+2)−7x=3x+6 (1 point)ResponsesThe equation has one solution: x=6.The equation has one solution: x equals 6 .The equation has infinite solutions.The equation has infinite solutions.The equation has no solution.The equation has no solution.The equation has one solution: x=10.The equation has one solution: x equals 10 .
Let's solve the equation step by step:
5(2x+2) - 7x = 3x + 6
Distribute the 5 on the left side:
10x + 10 - 7x = 3x + 6
Combine like terms:
3x + 10 = 3x + 6
Subtract 3x from both sides:
10 = 6
This equation is not true, as 10 is not equal to 6. So, there are no solutions to this equation.
To determine which of the given equations has no solution, you need to compare the coefficients and constants on both sides of the equation.
Let's analyze each equation:
1. 3x + 4 = 3x + 4
In this equation, the coefficients and constants on both sides are the same, indicating that the equation is consistent. Therefore, it has infinitely many solutions.
2. 3x + 4 = 4x + 3
Here, the coefficients on both sides are different, which means the equation has a unique solution. However, let's solve it to be sure:
Move all the terms with 'x' to one side:
3x - 4x = 3 - 4
-x = -1
Divide both sides by -1:
x = 1
So, this equation has a unique solution, x = 1.
3. 4x + 3 = 3x + 3
Similar to the previous equation, let's compare coefficients and constants:
4x - 3x = 3 - 3
x = 0
Thus, this equation also has a unique solution, x = 0.
4. 3x + 4 = 3x + 3
If we compare the coefficients and constants, we can see they are the same on both sides.
3x - 3x = 3 - 4
0 = -1
In this case, we obtain an inconsistent statement, 0 = -1, which is false. Therefore, this equation has no solution.
By analyzing the given equations, we find that the equation:
3x + 4 = 3x + 3
is an example of an equation with no solution.