Tell whether each equation has one solution, infinitely many solutions, or no solution.
a. 6x + 8 = 6(x + 2)
b. 10x = 15 + 5x
c. x + 11 = 8x + 11 - 7x
a. 6x+8 = 6(x+2)
6x+8 = 6x+12
8 = 12
There is no value of x which will make that true. So, no solution.
Now try the others. See whether there is a solution (solve normally), or whether the final sentence is true regardless of the value of x.
To determine whether each equation has one solution, infinitely many solutions, or no solution, we need to simplify and solve each equation. Let's go through each equation step by step:
a. 6x + 8 = 6(x + 2)
First, let's simplify the equation by distributing the 6:
6x + 8 = 6x + 12
Next, let's isolate the variable x by subtracting 6x from both sides:
8 = 12
This equation is false because 8 does not equal 12. Therefore, there is no solution for this equation.
b. 10x = 15 + 5x
To simplify the equation, let's combine like terms by subtracting 5x from both sides:
10x - 5x = 15
Simplifying further:
5x = 15
To isolate the variable x, we divide both sides by 5:
5x/5 = 15/5
x = 3
This equation has one solution, x = 3.
c. x + 11 = 8x + 11 - 7x
First, let's simplify the equation by combining like terms on the right side:
x + 11 = x + 11
Notice that the variable x cancels out on both sides:
0 = 0
This equation is always true. Therefore, there are infinitely many solutions for this equation.
In summary:
a. 6x + 8 = 6(x + 2) has no solution.
b. 10x = 15 + 5x has one solution, x = 3.
c. x + 11 = 8x + 11 - 7x has infinitely many solutions.