To determine if the equation [tex]\((7 + y)^2 - y^2 = 9x + 12\)[/tex] is linear and, if so, convert it to standard form [tex]\(ax + by = c\)[/tex], follow the steps outlined below:
1. Simplify the left-hand side:
Expand [tex]\((7 + y)^2\)[/tex]:
[tex]\[
(7 + y)^2 = 49 + 14y + y^2
\][/tex]
Subtract [tex]\(y^2\)[/tex] from the expanded form:
[tex]\[
49 + 14y + y^2 - y^2 = 49 + 14y
\][/tex]
Thus, the equation becomes:
[tex]\[
49 + 14y = 9x + 12
\][/tex]
2. Rearrange the equation to the form [tex]\(ax + by = c\)[/tex]:
Move the terms to one side to arrange the equation into the standard form. Subtract [tex]\(49\)[/tex] and [tex]\(14y\)[/tex] from both sides:
[tex]\[
49 + 14y - 49 - 14y = 9x + 12 - 49 - 14y
\][/tex]
Simplify the equation:
[tex]\[
0 = 9x - 14y + 12 - 49
\][/tex]
[tex]\[
0 = 9x - 14y - 37
\][/tex]
[tex]\[
9x - 14y = 37
\][/tex]
3. Conclusion:
The given equation is linear because it can be rearranged into the standard form [tex]\(ax + by = c\)[/tex] where [tex]\(a = 9\)[/tex], [tex]\(b = -14\)[/tex], and [tex]\(c = 37\)[/tex].
Thus, the standard form of the given linear equation is:
[tex]\[
9x - 14y = 37
\][/tex]
