Determine the missing values that would make this equation have infinite amount of solutions
2(6x+7)=3x+?x=?
To have an infinite amount of solutions, the equation must be equivalent to zero. Thus, we need to determine the values of "?" and "?" that make 2(6x + 7) = 3x + ?x equivalent to zero.
Expanding the left side of the equation, we get 12x + 14. Setting this equal to zero, we have 12x + 14 = 0. Simplifying, we subtract 14 from both sides to isolate the variable x: 12x = -14. Lastly, we divide both sides by 12 to solve for x: x = -14/12.
Therefore, no matter the values of "?" and "?", the equation can never be equivalent to zero and, thus, there are no missing values that would make it have an infinite amount of solutions.
To find the missing values that would make the equation have an infinite number of solutions, we need to find the values for "?" and "x" that would make both sides of the equation equal.
Given the equation:
2(6x + 7) = 3x + ?x
We can start by simplifying the left side of the equation:
2(6x + 7) = 12x + 14
Now we have the equation:
12x + 14 = 3x + ?x
To have an infinite number of solutions, we need to ensure that the two sides of the equation are equal regardless of the value of "x".
For this to happen, the coefficients of x on both sides of the equation must be the same. In this case, 12x and 3x have different coefficients (12 and 3).
To make the coefficients equal and have an infinite number of solutions, we need to choose "?" in a way that makes 3x + ?x equal to 12x.
So, we find the value of "?" by solving the equation 3x + ?x = 12x:
3x + ?x = 12x
(3 + ?)x = 12x
To make (3 + ?)x equal to 12x, the coefficient of x must be the same on both sides. Therefore, we can set (3 + ?) equal to 12:
3 + ? = 12
Solving for "?" gives us:
? = 12 - 3
? = 9
So, the missing value that would make this equation have an infinite number of solutions is "? = 9".
To determine the missing values that would make the equation have an infinite number of solutions, we need to solve the given equation.
First, let's simplify the equation by distributing the 2 to the terms within the parentheses:
2(6x + 7) = 3x + ?x
This gives us:
12x + 14 = 3x + ?x
Next, we want to combine like terms by transferring all the x terms to one side of the equation and the constants to the other side:
12x - 3x + 14 = ?x
This simplifies to:
9x + 14 = ?x
At this point, we can see that there is a missing value representing the coefficient of x, denoted as "?". In order for the equation to have an infinite number of solutions, the coefficient of x on one side of the equation should be equal to the coefficient of x on the other side. Therefore, the missing value ?x must be equal to 9.
So, the missing values are:
?x = 9
If we substitute this value back into the equation, we get:
9x + 14 = 9x
This equation has the same variable on both sides, which means that it is true for any value of x. Therefore, this equation has an infinite number of solutions.