To determine the value of [tex]\( k \)[/tex] such that the expression [tex]\( 5(k + x) + 7k \)[/tex] is equivalent to the expression [tex]\( 5x - 1 \)[/tex], we need to follow these steps:
1. Expand and Simplify the Left-Hand Side:
Start with the left-hand expression:
[tex]\[
5(k + x) + 7k
\][/tex]
Distribute the [tex]\( 5 \)[/tex] across the terms inside the parentheses:
[tex]\[
5k + 5x + 7k
\][/tex]
Combine the terms involving [tex]\( k \)[/tex]:
[tex]\[
5x + 12k
\][/tex]
2. Set the Expressions Equal to Each Other:
Now that we have simplified the left-hand side to [tex]\( 5x + 12k \)[/tex], we can set it equal to the right-hand side expression:
[tex]\[
5x + 12k = 5x - 1
\][/tex]
3. Isolate [tex]\( k \)[/tex]:
To solve for [tex]\( k \)[/tex], subtract [tex]\( 5x \)[/tex] from both sides of the equation:
[tex]\[
12k = -1
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
Divide both sides of the equation by 12 to isolate [tex]\( k \)[/tex]:
[tex]\[
k = \frac{-1}{12}
\][/tex]
Thus, the value of [tex]\( k \)[/tex] that satisfies the given equation is:
[tex]\[
k = -\frac{1}{12}
\][/tex]
