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x+9y=297,-2x+y=-138
To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.
x+9y=297
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=-9y+297
Subtract 9y from both sides of the equation.
-2\left(-9y+297\right)+y=-138
Substitute -9y+297 for x in the other equation, -2x+y=-138.
18y-594+y=-138
Multiply -2 times -9y+297.
19y-594=-138
Add 18y to y.
19y=456
Add 594 to both sides of the equation.
y=24
Divide both sides by 19.
x=-9\times 24+297
Substitute 24 for y in x=-9y+297. Because the resulting equation contains only one variable, you can solve for x directly.
x=-216+297
Multiply -9 times 24.
x=81
Add 297 to -216.
x=81,y=24
The system is now solved.
x+9y=297,-2x+y=-138
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&9\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}297\\-138\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&9\\-2&1\end{matrix}\right))\left(\begin{matrix}1&9\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&9\\-2&1\end{matrix}\right))\left(\begin{matrix}297\\-138\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&9\\-2&1\end{matrix}\right).
\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&9\\-2&1\end{matrix}\right))\left(\begin{matrix}297\\-138\end{matrix}\right)
The product of a matrix and its inverse is the identity matrix.
\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&9\\-2&1\end{matrix}\right))\left(\begin{matrix}297\\-138\end{matrix}\right)
Multiply the matrices on the left hand side of the equal sign.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-9\left(-2\right)}&-\frac{9}{1-9\left(-2\right)}\\-\frac{-2}{1-9\left(-2\right)}&\frac{1}{1-9\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}297\\-138\end{matrix}\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{19}&-\frac{9}{19}\\\frac{2}{19}&\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}297\\-138\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{19}\times 297-\frac{9}{19}\left(-138\right)\\\frac{2}{19}\times 297+\frac{1}{19}\left(-138\right)\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}81\\24\end{matrix}\right)
Do the arithmetic.
x=81,y=24
Extract the matrix elements x and y.
x+9y=297,-2x+y=-138
In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.
-2x-2\times 9y=-2\times 297,-2x+y=-138
To make x and -2x equal, multiply all terms on each side of the first equation by -2 and all terms on each side of the second by 1.
-2x-18y=-594,-2x+y=-138
Simplify.
-2x+2x-18y-y=-594+138
Subtract -2x+y=-138 from -2x-18y=-594 by subtracting like terms on each side of the equal sign.
-18y-y=-594+138
Add -2x to 2x. Terms -2x and 2x cancel out, leaving an equation with only one variable that can be solved.
-19y=-594+138
Add -18y to -y.
-19y=-456
Add -594 to 138.
y=24
Divide both sides by -19.
-2x+24=-138
Substitute 24 for y in -2x+y=-138. Because the resulting equation contains only one variable, you can solve for x directly.
-2x=-162
Subtract 24 from both sides of the equation.
x=81
Divide both sides by -2.
x=81,y=24
The system is now solved.