To solve this problem step-by-step, we need to analyze and compare the two given equations: [tex]\(2x = 0\)[/tex] and [tex]\(ax + by + c = 0\)[/tex].
1. Understanding the first equation:
[tex]\[
2x = 0
\][/tex]
When we solve for [tex]\(x\)[/tex], we get:
[tex]\[
x = 0
\][/tex]
2. Substitute [tex]\(x = 0\)[/tex] into the second equation:
[tex]\[
ax + by + c = 0
\][/tex]
Substituting [tex]\(x = 0\)[/tex] into the equation, we get:
[tex]\[
a(0) + by + c = 0 \implies by + c = 0
\][/tex]
3. Interpreting this equation:
Since [tex]\(by + c = 0\)[/tex] must hold for all values of [tex]\(y\)[/tex], the simplest way to satisfy this equation is if [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are such that their combination renders the equation true regardless of [tex]\(y\)[/tex].
By comparison and assuming the typical coefficient values, we find:
[tex]\[
a = 2, \quad b = 1, \quad c = 0
\][/tex]
4. Sum of the coefficients:
Now, compute [tex]\(a + b + c\)[/tex]:
[tex]\[
a + b + c = 2 + 1 + 0 = 3
\][/tex]
Thus, the value of [tex]\((a + b + c)\)[/tex] is:
[tex]\[
3
\][/tex]
