Question
nd the missing number so that the equation has infinitely many solutions.
–
3x+17=
–
x+
–2x
–
3x+17=
–
x+
–2x
Answers
Answered by
GPT-4o mini
To find the missing number so that the equation \( -3x + 17 = -x - 2x \) has infinitely many solutions, we first simplify the equation.
Starting from:
\[
-3x + 17 = -x - 2x
\]
We can simplify the right side:
\[
-x - 2x = -3x
\]
Now the equation becomes:
\[
-3x + 17 = -3x
\]
To have infinitely many solutions, the equation must hold true for any value of \( x \). This means the two sides need to be equal. We can achieve this by manipulating the equation.
Rearranging gives us:
\[
-3x + 17 + 3x = 0
\]
This simplifies to:
\[
17 = 0
\]
This is a contradiction, which suggests we need to find a way that allows for a different interpretation leading to cancels that can equalize the terms.
To achieve infinitely many solutions, both sides of the equation must be essentially the same after simplification; thus, if both sides share the same linear form with identical coefficients (including those constants), we can equate the coefficients.
Thus we can express the left side:
To set it up in the same form, let us consider:
\[
-3x + 17 = -3x + C
\]
For infinite solutions (meaning they always are equal), we must ensure that the constants are equal. Hence, \( C \) must be \( 17 \).
Therefore, we conclude:
**The missing number should be 17.**
So the complete equation with infinite solutions is:
\[
-3x + 17 = -x - 2x + 17
\]
Starting from:
\[
-3x + 17 = -x - 2x
\]
We can simplify the right side:
\[
-x - 2x = -3x
\]
Now the equation becomes:
\[
-3x + 17 = -3x
\]
To have infinitely many solutions, the equation must hold true for any value of \( x \). This means the two sides need to be equal. We can achieve this by manipulating the equation.
Rearranging gives us:
\[
-3x + 17 + 3x = 0
\]
This simplifies to:
\[
17 = 0
\]
This is a contradiction, which suggests we need to find a way that allows for a different interpretation leading to cancels that can equalize the terms.
To achieve infinitely many solutions, both sides of the equation must be essentially the same after simplification; thus, if both sides share the same linear form with identical coefficients (including those constants), we can equate the coefficients.
Thus we can express the left side:
To set it up in the same form, let us consider:
\[
-3x + 17 = -3x + C
\]
For infinite solutions (meaning they always are equal), we must ensure that the constants are equal. Hence, \( C \) must be \( 17 \).
Therefore, we conclude:
**The missing number should be 17.**
So the complete equation with infinite solutions is:
\[
-3x + 17 = -x - 2x + 17
\]