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2x+5y=130,4x+3y=218
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.
2x+5y=130
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
2x=-5y+130
Subtract 5y from both sides of the equation.
x=\frac{1}{2}\left(-5y+130\right)
Divide both sides by 2.
x=-\frac{5}{2}y+65
Multiply \frac{1}{2} times -5y+130.
4\left(-\frac{5}{2}y+65\right)+3y=218
Substitute -\frac{5y}{2}+65 for x in the other equation, 4x+3y=218.
-10y+260+3y=218
Multiply 4 times -\frac{5y}{2}+65.
-7y+260=218
Add -10y to 3y.
-7y=-42
Subtract 260 from both sides of the equation.
y=6
Divide both sides by -7.
x=-\frac{5}{2}\times 6+65
Substitute 6 for y in x=-\frac{5}{2}y+65. Because the resulting equation contains only one variable, you can solve for x directly.
x=-15+65
Multiply -\frac{5}{2} times 6.
x=50
Add 65 to -15.
x=50,y=6
The system is now solved.
2x+5y=130,4x+3y=218
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}2&5\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}130\\218\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}2&5\\4&3\end{matrix}\right))\left(\begin{matrix}2&5\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\4&3\end{matrix}\right))\left(\begin{matrix}130\\218\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&5\\4&3\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}2&5\\4&3\end{matrix}\right))\left(\begin{matrix}130\\218\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}2&5\\4&3\end{matrix}\right))\left(\begin{matrix}130\\218\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{3}{2\times 3-5\times 4}&-\frac{5}{2\times 3-5\times 4}\\-\frac{4}{2\times 3-5\times 4}&\frac{2}{2\times 3-5\times 4}\end{matrix}\right)\left(\begin{matrix}130\\218\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{3}{14}&\frac{5}{14}\\\frac{2}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}130\\218\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{14}\times 130+\frac{5}{14}\times 218\\\frac{2}{7}\times 130-\frac{1}{7}\times 218\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\6\end{matrix}\right)
Do the arithmetic.
x=50,y=6
Extract the matrix elements x and y.
2x+5y=130,4x+3y=218
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.
4\times 2x+4\times 5y=4\times 130,2\times 4x+2\times 3y=2\times 218
To make 2x and 4x equal, multiply all terms on each side of the first equation by 4 and all terms on each side of the second by 2.
8x+20y=520,8x+6y=436
Simplify.
8x-8x+20y-6y=520-436
Subtract 8x+6y=436 from 8x+20y=520 by subtracting like terms on each side of the equal sign.
20y-6y=520-436
Add 8x to -8x. Terms 8x and -8x cancel out, leaving an equation with only one variable that can be solved.
14y=520-436
Add 20y to -6y.
14y=84
Add 520 to -436.
y=6
Divide both sides by 14.
4x+3\times 6=218
Substitute 6 for y in 4x+3y=218. Because the resulting equation contains only one variable, you can solve for x directly.
4x+18=218
Multiply 3 times 6.
4x=200
Subtract 18 from both sides of the equation.
x=50
Divide both sides by 4.
x=50,y=6
The system is now solved.