\left\{ \begin{array} { l } { \frac { 3 x + 5 y } { 7 } - \frac { 4 x - 2 y } { 3 } = \frac { 71 } { 14 } } \\ { \frac { 2 x - 4 y } { 5 } + \frac { y - 6 x } { 2 } = - \frac { 5 } { 2 } } \end{array} \right.
Solve for x, y
x=\frac{1}{2}=0.5
y=4
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6\left(3x+5y\right)-14\left(4x-2y\right)=213
Consider the first equation. Multiply both sides of the equation by 42, the least common multiple of 7,3,14.
18x+30y-14\left(4x-2y\right)=213
Use the distributive property to multiply 6 by 3x+5y.
18x+30y-56x+28y=213
Use the distributive property to multiply -14 by 4x-2y.
-38x+30y+28y=213
Combine 18x and -56x to get -38x.
-38x+58y=213
Combine 30y and 28y to get 58y.
2\left(2x-4y\right)+5\left(y-6x\right)=-25
Consider the second equation. Multiply both sides of the equation by 10, the least common multiple of 5,2.
4x-8y+5\left(y-6x\right)=-25
Use the distributive property to multiply 2 by 2x-4y.
4x-8y+5y-30x=-25
Use the distributive property to multiply 5 by y-6x.
4x-3y-30x=-25
Combine -8y and 5y to get -3y.
-26x-3y=-25
Combine 4x and -30x to get -26x.
-38x+58y=213,-26x-3y=-25
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.
-38x+58y=213
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
-38x=-58y+213
Subtract 58y from both sides of the equation.
x=-\frac{1}{38}\left(-58y+213\right)
Divide both sides by -38.
x=\frac{29}{19}y-\frac{213}{38}
Multiply -\frac{1}{38} times -58y+213.
-26\left(\frac{29}{19}y-\frac{213}{38}\right)-3y=-25
Substitute \frac{29y}{19}-\frac{213}{38} for x in the other equation, -26x-3y=-25.
-\frac{754}{19}y+\frac{2769}{19}-3y=-25
Multiply -26 times \frac{29y}{19}-\frac{213}{38}.
-\frac{811}{19}y+\frac{2769}{19}=-25
Add -\frac{754y}{19} to -3y.
-\frac{811}{19}y=-\frac{3244}{19}
Subtract \frac{2769}{19} from both sides of the equation.
y=4
Divide both sides of the equation by -\frac{811}{19}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=\frac{29}{19}\times 4-\frac{213}{38}
Substitute 4 for y in x=\frac{29}{19}y-\frac{213}{38}. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{116}{19}-\frac{213}{38}
Multiply \frac{29}{19} times 4.
x=\frac{1}{2}
Add -\frac{213}{38} to \frac{116}{19} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{1}{2},y=4
The system is now solved.
6\left(3x+5y\right)-14\left(4x-2y\right)=213
Consider the first equation. Multiply both sides of the equation by 42, the least common multiple of 7,3,14.
18x+30y-14\left(4x-2y\right)=213
Use the distributive property to multiply 6 by 3x+5y.
18x+30y-56x+28y=213
Use the distributive property to multiply -14 by 4x-2y.
-38x+30y+28y=213
Combine 18x and -56x to get -38x.
-38x+58y=213
Combine 30y and 28y to get 58y.
2\left(2x-4y\right)+5\left(y-6x\right)=-25
Consider the second equation. Multiply both sides of the equation by 10, the least common multiple of 5,2.
4x-8y+5\left(y-6x\right)=-25
Use the distributive property to multiply 2 by 2x-4y.
4x-8y+5y-30x=-25
Use the distributive property to multiply 5 by y-6x.
4x-3y-30x=-25
Combine -8y and 5y to get -3y.
-26x-3y=-25
Combine 4x and -30x to get -26x.
-38x+58y=213,-26x-3y=-25
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}-38&58\\-26&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}213\\-25\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}-38&58\\-26&-3\end{matrix}\right))\left(\begin{matrix}-38&58\\-26&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-38&58\\-26&-3\end{matrix}\right))\left(\begin{matrix}213\\-25\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}-38&58\\-26&-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}-38&58\\-26&-3\end{matrix}\right))\left(\begin{matrix}213\\-25\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}-38&58\\-26&-3\end{matrix}\right))\left(\begin{matrix}213\\-25\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}{-38\left(-3\right)-58\left(-26\right)}&-\frac{58}{-38\left(-3\right)-58\left(-26\right)}\\-\frac{-26}{-38\left(-3\right)-58\left(-26\right)}&-\frac{38}{-38\left(-3\right)-58\left(-26\right)}\end{matrix}\right)\left(\begin{matrix}213\\-25\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}{1622}&-\frac{29}{811}\\\frac{13}{811}&-\frac{19}{811}\end{matrix}\right)\left(\begin{matrix}213\\-25\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{1622}\times 213-\frac{29}{811}\left(-25\right)\\\frac{13}{811}\times 213-\frac{19}{811}\left(-25\right)\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\\4\end{matrix}\right)
Do the arithmetic.
x=\frac{1}{2},y=4
Extract the matrix elements x and y.
6\left(3x+5y\right)-14\left(4x-2y\right)=213
Consider the first equation. Multiply both sides of the equation by 42, the least common multiple of 7,3,14.
18x+30y-14\left(4x-2y\right)=213
Use the distributive property to multiply 6 by 3x+5y.
18x+30y-56x+28y=213
Use the distributive property to multiply -14 by 4x-2y.
-38x+30y+28y=213
Combine 18x and -56x to get -38x.
-38x+58y=213
Combine 30y and 28y to get 58y.
2\left(2x-4y\right)+5\left(y-6x\right)=-25
Consider the second equation. Multiply both sides of the equation by 10, the least common multiple of 5,2.
4x-8y+5\left(y-6x\right)=-25
Use the distributive property to multiply 2 by 2x-4y.
4x-8y+5y-30x=-25
Use the distributive property to multiply 5 by y-6x.
4x-3y-30x=-25
Combine -8y and 5y to get -3y.
-26x-3y=-25
Combine 4x and -30x to get -26x.
-38x+58y=213,-26x-3y=-25
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.
-26\left(-38\right)x-26\times 58y=-26\times 213,-38\left(-26\right)x-38\left(-3\right)y=-38\left(-25\right)
To make -38x and -26x equal, multiply all terms on each side of the first equation by -26 and all terms on each side of the second by -38.
988x-1508y=-5538,988x+114y=950
Simplify.
988x-988x-1508y-114y=-5538-950
Subtract 988x+114y=950 from 988x-1508y=-5538 by subtracting like terms on each side of the equal sign.
-1508y-114y=-5538-950
Add 988x to -988x. Terms 988x and -988x cancel out, leaving an equation with only one variable that can be solved.
-1622y=-5538-950
Add -1508y to -114y.
-1622y=-6488
Add -5538 to -950.
y=4
Divide both sides by -1622.
-26x-3\times 4=-25
Substitute 4 for y in -26x-3y=-25. Because the resulting equation contains only one variable, you can solve for x directly.
-26x-12=-25
Multiply -3 times 4.
-26x=-13
Add 12 to both sides of the equation.
x=\frac{1}{2}
Divide both sides by -26.
x=\frac{1}{2},y=4
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}