\left\{ \begin{array} { l } { x - \frac { 3 x + 4 } { 7 } = \frac { y + 2 } { 3 } } \\ { 2 y - \frac { 5 x + 4 } { 11 } = \frac { x + 24 } { 2 } } \end{array} \right.
Solve for x, y
x=8
y=10
Graph
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21x-3\left(3x+4\right)=7\left(y+2\right)
Consider the first equation. Multiply both sides of the equation by 21, the least common multiple of 7,3.
21x-9x-12=7\left(y+2\right)
Use the distributive property to multiply -3 by 3x+4.
12x-12=7\left(y+2\right)
Combine 21x and -9x to get 12x.
12x-12=7y+14
Use the distributive property to multiply 7 by y+2.
12x-12-7y=14
Subtract 7y from both sides.
12x-7y=14+12
Add 12 to both sides.
12x-7y=26
Add 14 and 12 to get 26.
44y-2\left(5x+4\right)=11\left(x+24\right)
Consider the second equation. Multiply both sides of the equation by 22, the least common multiple of 11,2.
44y-10x-8=11\left(x+24\right)
Use the distributive property to multiply -2 by 5x+4.
44y-10x-8=11x+264
Use the distributive property to multiply 11 by x+24.
44y-10x-8-11x=264
Subtract 11x from both sides.
44y-21x-8=264
Combine -10x and -11x to get -21x.
44y-21x=264+8
Add 8 to both sides.
44y-21x=272
Add 264 and 8 to get 272.
12x-7y=26,-21x+44y=272
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.
12x-7y=26
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
12x=7y+26
Add 7y to both sides of the equation.
x=\frac{1}{12}\left(7y+26\right)
Divide both sides by 12.
x=\frac{7}{12}y+\frac{13}{6}
Multiply \frac{1}{12} times 7y+26.
-21\left(\frac{7}{12}y+\frac{13}{6}\right)+44y=272
Substitute \frac{7y}{12}+\frac{13}{6} for x in the other equation, -21x+44y=272.
-\frac{49}{4}y-\frac{91}{2}+44y=272
Multiply -21 times \frac{7y}{12}+\frac{13}{6}.
\frac{127}{4}y-\frac{91}{2}=272
Add -\frac{49y}{4} to 44y.
\frac{127}{4}y=\frac{635}{2}
Add \frac{91}{2} to both sides of the equation.
y=10
Divide both sides of the equation by \frac{127}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=\frac{7}{12}\times 10+\frac{13}{6}
Substitute 10 for y in x=\frac{7}{12}y+\frac{13}{6}. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{35+13}{6}
Multiply \frac{7}{12} times 10.
x=8
Add \frac{13}{6} to \frac{35}{6} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=8,y=10
The system is now solved.
21x-3\left(3x+4\right)=7\left(y+2\right)
Consider the first equation. Multiply both sides of the equation by 21, the least common multiple of 7,3.
21x-9x-12=7\left(y+2\right)
Use the distributive property to multiply -3 by 3x+4.
12x-12=7\left(y+2\right)
Combine 21x and -9x to get 12x.
12x-12=7y+14
Use the distributive property to multiply 7 by y+2.
12x-12-7y=14
Subtract 7y from both sides.
12x-7y=14+12
Add 12 to both sides.
12x-7y=26
Add 14 and 12 to get 26.
44y-2\left(5x+4\right)=11\left(x+24\right)
Consider the second equation. Multiply both sides of the equation by 22, the least common multiple of 11,2.
44y-10x-8=11\left(x+24\right)
Use the distributive property to multiply -2 by 5x+4.
44y-10x-8=11x+264
Use the distributive property to multiply 11 by x+24.
44y-10x-8-11x=264
Subtract 11x from both sides.
44y-21x-8=264
Combine -10x and -11x to get -21x.
44y-21x=264+8
Add 8 to both sides.
44y-21x=272
Add 264 and 8 to get 272.
12x-7y=26,-21x+44y=272
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}12&-7\\-21&44\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}26\\272\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}12&-7\\-21&44\end{matrix}\right))\left(\begin{matrix}12&-7\\-21&44\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&-7\\-21&44\end{matrix}\right))\left(\begin{matrix}26\\272\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}12&-7\\-21&44\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}12&-7\\-21&44\end{matrix}\right))\left(\begin{matrix}26\\272\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}12&-7\\-21&44\end{matrix}\right))\left(\begin{matrix}26\\272\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{44}{12\times 44-\left(-7\left(-21\right)\right)}&-\frac{-7}{12\times 44-\left(-7\left(-21\right)\right)}\\-\frac{-21}{12\times 44-\left(-7\left(-21\right)\right)}&\frac{12}{12\times 44-\left(-7\left(-21\right)\right)}\end{matrix}\right)\left(\begin{matrix}26\\272\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{44}{381}&\frac{7}{381}\\\frac{7}{127}&\frac{4}{127}\end{matrix}\right)\left(\begin{matrix}26\\272\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{44}{381}\times 26+\frac{7}{381}\times 272\\\frac{7}{127}\times 26+\frac{4}{127}\times 272\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\10\end{matrix}\right)
Do the arithmetic.
x=8,y=10
Extract the matrix elements x and y.
21x-3\left(3x+4\right)=7\left(y+2\right)
Consider the first equation. Multiply both sides of the equation by 21, the least common multiple of 7,3.
21x-9x-12=7\left(y+2\right)
Use the distributive property to multiply -3 by 3x+4.
12x-12=7\left(y+2\right)
Combine 21x and -9x to get 12x.
12x-12=7y+14
Use the distributive property to multiply 7 by y+2.
12x-12-7y=14
Subtract 7y from both sides.
12x-7y=14+12
Add 12 to both sides.
12x-7y=26
Add 14 and 12 to get 26.
44y-2\left(5x+4\right)=11\left(x+24\right)
Consider the second equation. Multiply both sides of the equation by 22, the least common multiple of 11,2.
44y-10x-8=11\left(x+24\right)
Use the distributive property to multiply -2 by 5x+4.
44y-10x-8=11x+264
Use the distributive property to multiply 11 by x+24.
44y-10x-8-11x=264
Subtract 11x from both sides.
44y-21x-8=264
Combine -10x and -11x to get -21x.
44y-21x=264+8
Add 8 to both sides.
44y-21x=272
Add 264 and 8 to get 272.
12x-7y=26,-21x+44y=272
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.
-21\times 12x-21\left(-7\right)y=-21\times 26,12\left(-21\right)x+12\times 44y=12\times 272
To make 12x and -21x equal, multiply all terms on each side of the first equation by -21 and all terms on each side of the second by 12.
-252x+147y=-546,-252x+528y=3264
Simplify.
-252x+252x+147y-528y=-546-3264
Subtract -252x+528y=3264 from -252x+147y=-546 by subtracting like terms on each side of the equal sign.
147y-528y=-546-3264
Add -252x to 252x. Terms -252x and 252x cancel out, leaving an equation with only one variable that can be solved.
-381y=-546-3264
Add 147y to -528y.
-381y=-3810
Add -546 to -3264.
y=10
Divide both sides by -381.
-21x+44\times 10=272
Substitute 10 for y in -21x+44y=272. Because the resulting equation contains only one variable, you can solve for x directly.
-21x+440=272
Multiply 44 times 10.
-21x=-168
Subtract 440 from both sides of the equation.
x=8
Divide both sides by -21.
x=8,y=10
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}