\left. \begin{array} { l l } { - 9 x - 9 z = 36 } \\ { 9 x + 2 z = 72 } \end{array} \right.
Solve for x, z
x = \frac{80}{7} = 11\frac{3}{7} \approx 11.428571429
z = -\frac{108}{7} = -15\frac{3}{7} \approx -15.428571429
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-9x-9z=36,9x+2z=72
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.
-9x-9z=36
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
-9x=9z+36
Add 9z to both sides of the equation.
x=-\frac{1}{9}\left(9z+36\right)
Divide both sides by -9.
x=-z-4
Multiply -\frac{1}{9} times 36+9z.
9\left(-z-4\right)+2z=72
Substitute -z-4 for x in the other equation, 9x+2z=72.
-9z-36+2z=72
Multiply 9 times -z-4.
-7z-36=72
Add -9z to 2z.
-7z=108
Add 36 to both sides of the equation.
z=-\frac{108}{7}
Divide both sides by -7.
x=-\left(-\frac{108}{7}\right)-4
Substitute -\frac{108}{7} for z in x=-z-4. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{108}{7}-4
Multiply -1 times -\frac{108}{7}.
x=\frac{80}{7}
Add -4 to \frac{108}{7}.
x=\frac{80}{7},z=-\frac{108}{7}
The system is now solved.
-9x-9z=36,9x+2z=72
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}-9&-9\\9&2\end{matrix}\right)\left(\begin{matrix}x\\z\end{matrix}\right)=\left(\begin{matrix}36\\72\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}-9&-9\\9&2\end{matrix}\right))\left(\begin{matrix}-9&-9\\9&2\end{matrix}\right)\left(\begin{matrix}x\\z\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-9\\9&2\end{matrix}\right))\left(\begin{matrix}36\\72\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}-9&-9\\9&2\end{matrix}\right).
\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\z\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-9\\9&2\end{matrix}\right))\left(\begin{matrix}36\\72\end{matrix}\right)
The product of a matrix and its inverse is the identity matrix.
\left(\begin{matrix}x\\z\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-9\\9&2\end{matrix}\right))\left(\begin{matrix}36\\72\end{matrix}\right)
Multiply the matrices on the left hand side of the equal sign.
\left(\begin{matrix}x\\z\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-9\times 2-\left(-9\times 9\right)}&-\frac{-9}{-9\times 2-\left(-9\times 9\right)}\\-\frac{9}{-9\times 2-\left(-9\times 9\right)}&-\frac{9}{-9\times 2-\left(-9\times 9\right)}\end{matrix}\right)\left(\begin{matrix}36\\72\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\\z\end{matrix}\right)=\left(\begin{matrix}\frac{2}{63}&\frac{1}{7}\\-\frac{1}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}36\\72\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\z\end{matrix}\right)=\left(\begin{matrix}\frac{2}{63}\times 36+\frac{1}{7}\times 72\\-\frac{1}{7}\times 36-\frac{1}{7}\times 72\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\z\end{matrix}\right)=\left(\begin{matrix}\frac{80}{7}\\-\frac{108}{7}\end{matrix}\right)
Do the arithmetic.
x=\frac{80}{7},z=-\frac{108}{7}
Extract the matrix elements x and z.
-9x-9z=36,9x+2z=72
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.
9\left(-9\right)x+9\left(-9\right)z=9\times 36,-9\times 9x-9\times 2z=-9\times 72
To make -9x and 9x equal, multiply all terms on each side of the first equation by 9 and all terms on each side of the second by -9.
-81x-81z=324,-81x-18z=-648
Simplify.
-81x+81x-81z+18z=324+648
Subtract -81x-18z=-648 from -81x-81z=324 by subtracting like terms on each side of the equal sign.
-81z+18z=324+648
Add -81x to 81x. Terms -81x and 81x cancel out, leaving an equation with only one variable that can be solved.
-63z=324+648
Add -81z to 18z.
-63z=972
Add 324 to 648.
z=-\frac{108}{7}
Divide both sides by -63.
9x+2\left(-\frac{108}{7}\right)=72
Substitute -\frac{108}{7} for z in 9x+2z=72. Because the resulting equation contains only one variable, you can solve for x directly.
9x-\frac{216}{7}=72
Multiply 2 times -\frac{108}{7}.
9x=\frac{720}{7}
Add \frac{216}{7} to both sides of the equation.
x=\frac{80}{7}
Divide both sides by 9.
x=\frac{80}{7},z=-\frac{108}{7}
The system is now solved.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}