To solve for [tex]\( x \)[/tex] so that [tex]\( A \parallel B \)[/tex], we need to understand that parallel lines have the same slope.
Let's go through this step-by-step:
1. Identify the equations:
- The equation for line [tex]\( A \)[/tex] is given as [tex]\( 3x + 20 \)[/tex].
2. Equation for line [tex]\( B \)[/tex]:
- Since we need line [tex]\( B \)[/tex] to be parallel to line [tex]\( A \)[/tex], and parallel lines have the same slope, the general form for line [tex]\( B \)[/tex] would also have the same slope as line [tex]\( A \)[/tex], but possibly a different y-intercept. Thus, if we denote the line [tex]\( B \)[/tex] in the form [tex]\( 3x + k \)[/tex].
3. Setting up the equation:
- To solve for [tex]\( x \)[/tex], we need to set the equation of [tex]\( A \)[/tex] equal to the equation of [tex]\( B \)[/tex]. Since both lines are parallel, their equations would intersect when they are set equal to each other:
[tex]\[
3x + 20 = 3x + k
\][/tex]
4. Solving for [tex]\( x \)[/tex]:
- Simplifying the equation [tex]\( 3x + 20 = 3x + k \)[/tex]:
[tex]\[
\begin{align*}
3x + 20 &= 3x + k \\
20 &= k
\end{align*}
\][/tex]
- The above equation holds true regardless of [tex]\( x \)[/tex]. To find a specific value for [tex]\( x \)[/tex], we rearrange the equation:
[tex]\[
3x = k - 20
\][/tex]
[tex]\[
x = \frac{k - 20}{3}
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] that makes lines [tex]\( A \)[/tex] and [tex]\( B \)[/tex] parallel is:
[tex]\[
x = \frac{k}{3} - \frac{20}{3}.
\][/tex]
