How Many Solutions? Quick Check
5 of 55 of 5 Items
Question
Solve for x to determine if the equation has one, none, or infinite solutions.
5(2x+2)−7x=3x+6(1 point)
Responses
The equation has no solution.
The equation has no solution.
The equation has infinite solutions.
The equation has infinite solutions.
The equation has one solution: x=10.
The equation has one solution: x equals 10 .
The equation has one solution: x=6.
The equation has one solution: x equals 6 .
Solving the system y=−x+3 and x+y=8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain. (1 point) Responses The statement is true, because the lines have an intersection point. The statement is true, because the lines have an intersection point. The statement is false, because the lines are parallel. The statement is false, because the lines are parallel. The statement is true, because the lines are parallel. The statement is true, because the lines are parallel. The statement is false, because the lines have an intersection point. The statement is false, because the lines have an intersection point. Skip to navigation page 20 of 21
To solve for x and determine the number of solutions, we need to simplify the equation:
5(2x+2) - 7x = 3x + 6
First, let's distribute the 5 on the left side:
10x + 10 - 7x = 3x + 6
Combining like terms, we get:
3x + 10 = 3x + 6
Next, let's isolate the x terms on one side:
3x - 3x = 6 - 10
0 = -4
Since 0 is not equal to -4, this equation has no solution.
Therefore, the correct response is: The equation has no solution.
To determine how many solutions the equation has, let's solve it step by step.
Given equation: 5(2x + 2) - 7x = 3x + 6
1. Distribute the 5 to both terms inside the parentheses:
10x + 10 - 7x = 3x + 6
2. Combine like terms on both sides:
(10x - 7x) + 10 = (3x) + 6
3x + 10 = 3x + 6
3. Now, subtract 3x from both sides to isolate the constant terms:
3x - 3x + 10 = 3x - 3x + 6
10 = 6
4. As the equation simplifies to a false statement (10 = 6), there is no solution to this equation.
Therefore, the correct response is: The equation has no solution.