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3\times 3\left(x+2x+2\right)=4\left(4x+y-1\right)
Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.
9\left(x+2x+2\right)=4\left(4x+y-1\right)
Multiply 3 and 3 to get 9.
9\left(3x+2\right)=4\left(4x+y-1\right)
Combine x and 2x to get 3x.
27x+18=4\left(4x+y-1\right)
Use the distributive property to multiply 9 by 3x+2.
27x+18=16x+4y-4
Use the distributive property to multiply 4 by 4x+y-1.
27x+18-16x=4y-4
Subtract 16x from both sides.
11x+18=4y-4
Combine 27x and -16x to get 11x.
11x+18-4y=-4
Subtract 4y from both sides.
11x-4y=-4-18
Subtract 18 from both sides.
11x-4y=-22
Subtract 18 from -4 to get -22.
3\left(x+y\right)-\left(x-y\right)=y-1
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 2,6.
3x+3y-\left(x-y\right)=y-1
Use the distributive property to multiply 3 by x+y.
3x+3y-x+y=y-1
To find the opposite of x-y, find the opposite of each term.
2x+3y+y=y-1
Combine 3x and -x to get 2x.
2x+4y=y-1
Combine 3y and y to get 4y.
2x+4y-y=-1
Subtract y from both sides.
2x+3y=-1
Combine 4y and -y to get 3y.
11x-4y=-22,2x+3y=-1
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.
11x-4y=-22
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
11x=4y-22
Add 4y to both sides of the equation.
x=\frac{1}{11}\left(4y-22\right)
Divide both sides by 11.
x=\frac{4}{11}y-2
Multiply \frac{1}{11} times 4y-22.
2\left(\frac{4}{11}y-2\right)+3y=-1
Substitute \frac{4y}{11}-2 for x in the other equation, 2x+3y=-1.
\frac{8}{11}y-4+3y=-1
Multiply 2 times \frac{4y}{11}-2.
\frac{41}{11}y-4=-1
Add \frac{8y}{11} to 3y.
\frac{41}{11}y=3
Add 4 to both sides of the equation.
y=\frac{33}{41}
Divide both sides of the equation by \frac{41}{11}, which is the same as multiplying both sides by the reciprocal of the fraction.
x=\frac{4}{11}\times \frac{33}{41}-2
Substitute \frac{33}{41} for y in x=\frac{4}{11}y-2. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{12}{41}-2
Multiply \frac{4}{11} times \frac{33}{41} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=-\frac{70}{41}
Add -2 to \frac{12}{41}.
x=-\frac{70}{41},y=\frac{33}{41}
The system is now solved.
3\times 3\left(x+2x+2\right)=4\left(4x+y-1\right)
Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.
9\left(x+2x+2\right)=4\left(4x+y-1\right)
Multiply 3 and 3 to get 9.
9\left(3x+2\right)=4\left(4x+y-1\right)
Combine x and 2x to get 3x.
27x+18=4\left(4x+y-1\right)
Use the distributive property to multiply 9 by 3x+2.
27x+18=16x+4y-4
Use the distributive property to multiply 4 by 4x+y-1.
27x+18-16x=4y-4
Subtract 16x from both sides.
11x+18=4y-4
Combine 27x and -16x to get 11x.
11x+18-4y=-4
Subtract 4y from both sides.
11x-4y=-4-18
Subtract 18 from both sides.
11x-4y=-22
Subtract 18 from -4 to get -22.
3\left(x+y\right)-\left(x-y\right)=y-1
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 2,6.
3x+3y-\left(x-y\right)=y-1
Use the distributive property to multiply 3 by x+y.
3x+3y-x+y=y-1
To find the opposite of x-y, find the opposite of each term.
2x+3y+y=y-1
Combine 3x and -x to get 2x.
2x+4y=y-1
Combine 3y and y to get 4y.
2x+4y-y=-1
Subtract y from both sides.
2x+3y=-1
Combine 4y and -y to get 3y.
11x-4y=-22,2x+3y=-1
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}11&-4\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-22\\-1\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}11&-4\\2&3\end{matrix}\right))\left(\begin{matrix}11&-4\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}11&-4\\2&3\end{matrix}\right))\left(\begin{matrix}-22\\-1\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}11&-4\\2&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}11&-4\\2&3\end{matrix}\right))\left(\begin{matrix}-22\\-1\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}11&-4\\2&3\end{matrix}\right))\left(\begin{matrix}-22\\-1\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}{11\times 3-\left(-4\times 2\right)}&-\frac{-4}{11\times 3-\left(-4\times 2\right)}\\-\frac{2}{11\times 3-\left(-4\times 2\right)}&\frac{11}{11\times 3-\left(-4\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-22\\-1\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}{41}&\frac{4}{41}\\-\frac{2}{41}&\frac{11}{41}\end{matrix}\right)\left(\begin{matrix}-22\\-1\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{41}\left(-22\right)+\frac{4}{41}\left(-1\right)\\-\frac{2}{41}\left(-22\right)+\frac{11}{41}\left(-1\right)\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{70}{41}\\\frac{33}{41}\end{matrix}\right)
Do the arithmetic.
x=-\frac{70}{41},y=\frac{33}{41}
Extract the matrix elements x and y.
3\times 3\left(x+2x+2\right)=4\left(4x+y-1\right)
Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.
9\left(x+2x+2\right)=4\left(4x+y-1\right)
Multiply 3 and 3 to get 9.
9\left(3x+2\right)=4\left(4x+y-1\right)
Combine x and 2x to get 3x.
27x+18=4\left(4x+y-1\right)
Use the distributive property to multiply 9 by 3x+2.
27x+18=16x+4y-4
Use the distributive property to multiply 4 by 4x+y-1.
27x+18-16x=4y-4
Subtract 16x from both sides.
11x+18=4y-4
Combine 27x and -16x to get 11x.
11x+18-4y=-4
Subtract 4y from both sides.
11x-4y=-4-18
Subtract 18 from both sides.
11x-4y=-22
Subtract 18 from -4 to get -22.
3\left(x+y\right)-\left(x-y\right)=y-1
Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 2,6.
3x+3y-\left(x-y\right)=y-1
Use the distributive property to multiply 3 by x+y.
3x+3y-x+y=y-1
To find the opposite of x-y, find the opposite of each term.
2x+3y+y=y-1
Combine 3x and -x to get 2x.
2x+4y=y-1
Combine 3y and y to get 4y.
2x+4y-y=-1
Subtract y from both sides.
2x+3y=-1
Combine 4y and -y to get 3y.
11x-4y=-22,2x+3y=-1
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.
2\times 11x+2\left(-4\right)y=2\left(-22\right),11\times 2x+11\times 3y=11\left(-1\right)
To make 11x and 2x equal, multiply all terms on each side of the first equation by 2 and all terms on each side of the second by 11.
22x-8y=-44,22x+33y=-11
Simplify.
22x-22x-8y-33y=-44+11
Subtract 22x+33y=-11 from 22x-8y=-44 by subtracting like terms on each side of the equal sign.
-8y-33y=-44+11
Add 22x to -22x. Terms 22x and -22x cancel out, leaving an equation with only one variable that can be solved.
-41y=-44+11
Add -8y to -33y.
-41y=-33
Add -44 to 11.
y=\frac{33}{41}
Divide both sides by -41.
2x+3\times \frac{33}{41}=-1
Substitute \frac{33}{41} for y in 2x+3y=-1. Because the resulting equation contains only one variable, you can solve for x directly.
2x+\frac{99}{41}=-1
Multiply 3 times \frac{33}{41}.
2x=-\frac{140}{41}
Subtract \frac{99}{41} from both sides of the equation.
x=-\frac{70}{41}
Divide both sides by 2.
x=-\frac{70}{41},y=\frac{33}{41}
The system is now solved.