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Sagot :
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]
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]
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