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-7x+8y=-5,14x+3y=-36
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.
-7x+8y=-5
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
-7x=-8y-5
Subtract 8y from both sides of the equation.
x=-\frac{1}{7}\left(-8y-5\right)
Divide both sides by -7.
x=\frac{8}{7}y+\frac{5}{7}
Multiply -\frac{1}{7} times -8y-5.
14\left(\frac{8}{7}y+\frac{5}{7}\right)+3y=-36
Substitute \frac{8y+5}{7} for x in the other equation, 14x+3y=-36.
16y+10+3y=-36
Multiply 14 times \frac{8y+5}{7}.
19y+10=-36
Add 16y to 3y.
19y=-46
Subtract 10 from both sides of the equation.
y=-\frac{46}{19}
Divide both sides by 19.
x=\frac{8}{7}\left(-\frac{46}{19}\right)+\frac{5}{7}
Substitute -\frac{46}{19} for y in x=\frac{8}{7}y+\frac{5}{7}. Because the resulting equation contains only one variable, you can solve for x directly.
x=-\frac{368}{133}+\frac{5}{7}
Multiply \frac{8}{7} times -\frac{46}{19} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=-\frac{39}{19}
Add \frac{5}{7} to -\frac{368}{133} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=-\frac{39}{19},y=-\frac{46}{19}
The system is now solved.
-7x+8y=-5,14x+3y=-36
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}-7&8\\14&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\-36\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}-7&8\\14&3\end{matrix}\right))\left(\begin{matrix}-7&8\\14&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&8\\14&3\end{matrix}\right))\left(\begin{matrix}-5\\-36\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}-7&8\\14&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}-7&8\\14&3\end{matrix}\right))\left(\begin{matrix}-5\\-36\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}-7&8\\14&3\end{matrix}\right))\left(\begin{matrix}-5\\-36\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}{-7\times 3-8\times 14}&-\frac{8}{-7\times 3-8\times 14}\\-\frac{14}{-7\times 3-8\times 14}&-\frac{7}{-7\times 3-8\times 14}\end{matrix}\right)\left(\begin{matrix}-5\\-36\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}{133}&\frac{8}{133}\\\frac{2}{19}&\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}-5\\-36\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{133}\left(-5\right)+\frac{8}{133}\left(-36\right)\\\frac{2}{19}\left(-5\right)+\frac{1}{19}\left(-36\right)\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{39}{19}\\-\frac{46}{19}\end{matrix}\right)
Do the arithmetic.
x=-\frac{39}{19},y=-\frac{46}{19}
Extract the matrix elements x and y.
-7x+8y=-5,14x+3y=-36
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.
14\left(-7\right)x+14\times 8y=14\left(-5\right),-7\times 14x-7\times 3y=-7\left(-36\right)
To make -7x and 14x equal, multiply all terms on each side of the first equation by 14 and all terms on each side of the second by -7.
-98x+112y=-70,-98x-21y=252
Simplify.
-98x+98x+112y+21y=-70-252
Subtract -98x-21y=252 from -98x+112y=-70 by subtracting like terms on each side of the equal sign.
112y+21y=-70-252
Add -98x to 98x. Terms -98x and 98x cancel out, leaving an equation with only one variable that can be solved.
133y=-70-252
Add 112y to 21y.
133y=-322
Add -70 to -252.
y=-\frac{46}{19}
Divide both sides by 133.
14x+3\left(-\frac{46}{19}\right)=-36
Substitute -\frac{46}{19} for y in 14x+3y=-36. Because the resulting equation contains only one variable, you can solve for x directly.
14x-\frac{138}{19}=-36
Multiply 3 times -\frac{46}{19}.
14x=-\frac{546}{19}
Add \frac{138}{19} to both sides of the equation.
x=-\frac{39}{19}
Divide both sides by 14.
x=-\frac{39}{19},y=-\frac{46}{19}
The system is now solved.