Solve for \(x\) in \(2y=x+5\).
Substitute \(x=2y-5\) into \({x}^{2}+xy=-1\).
Start with the original equation.
\[{x}^{2}+xy=-1\]
Let \(x=2y-5\).
\[{(2y-5)}^{2}+(2y-5)y=-1\]
Simplify.
\[{(2y-5)}^{2}+y(2y-5)=-1\]
Solve for \(y\) in \({(2y-5)}^{2}+y(2y-5)=-1\).
Solve for \(y\).
\[{(2y-5)}^{2}+y(2y-5)=-1\]
Expand.
\[{(2y)}^{2}-2\times 2y\times 5+{5}^{2}+2{y}^{2}-5y=-1\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[{2}^{2}{y}^{2}-2\times 2y\times 5+{5}^{2}+2{y}^{2}-5y=-1\]
Simplify \({2}^{2}\) to \(4\).
\[4{y}^{2}-2\times 2y\times 5+{5}^{2}+2{y}^{2}-5y=-1\]
Simplify \({5}^{2}\) to \(25\).
\[4{y}^{2}-2\times 2y\times 5+25+2{y}^{2}-5y=-1\]
Simplify \(2\times 2y\times 5\) to \(20y\).
\[4{y}^{2}-20y+25+2{y}^{2}-5y=-1\]
Simplify \(4{y}^{2}-20y+25+2{y}^{2}-5y\) to \(6{y}^{2}-25y+25\).
\[6{y}^{2}-25y+25=-1\]
Move all terms to one side.
\[6{y}^{2}-25y+25+1=0\]
Simplify \(6{y}^{2}-25y+25+1\) to \(6{y}^{2}-25y+26\).
\[6{y}^{2}-25y+26=0\]
Split the second term in \(6{y}^{2}-25y+26\) into two terms.
Multiply the coefficient of the first term by the constant term.
\[6\times 26=156\]
Ask: Which two numbers add up to \(-25\) and multiply to \(156\)?
Split \(-25y\) as the sum of \(-12y\) and \(-13y\).
\[6{y}^{2}-12y-13y+26\]
\[6{y}^{2}-12y-13y+26=0\]
Factor out common terms in the first two terms, then in the last two terms.
\[6y(y-2)-13(y-2)=0\]
Factor out the common term \(y-2\).
\[(y-2)(6y-13)=0\]
Solve for \(y\).
Ask: When will \((y-2)(6y-13)\) equal zero?
When \(y-2=0\) or \(6y-13=0\)
Solve each of the 2 equations above.
\[y=2,\frac{13}{6}\]
\[y=2,\frac{13}{6}\]
Substitute \(y=2,\frac{13}{6}\) into \(x=2y-5\).
Start with the original equation.
\[x=2y-5\]
Let \(y=2,\frac{13}{6}\).
\[x=2\times 2-5,2\times \frac{13}{6}-5\]
Simplify.
\[x=-1,-\frac{2}{3}\]
