Solve {l}{2x+5y=-11}{2ax-5by=41}

$$\left\{ \begin{array} { l } { 2 x + 5 y = - 11 } \\ { 2 a x - 5 b y = 41 } \end{array} \right.$$

Solve for x, y (complex solution)

$$\left\{\begin{matrix}x=-\frac{11b-41}{2\left(a+b\right)}\text{, }y=-\frac{11a+41}{5\left(a+b\right)}\text{, }&a\neq -b\\x=\frac{-5y-11}{2}\text{, }y\in \mathrm{C}\text{, }&a=-\frac{41}{11}\text{ and }b=\frac{41}{11}\end{matrix}\right.$$

Steps Using Substitution

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.

$$2x+5y=-11,2ax+\left(-5b\right)y=41$$

Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.

$$2x+5y=-11$$

Subtract 5y from both sides of the equation.

$$2x=-5y-11$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(-5y-11\right)$$

Multiply \frac{1}{2} times -5y-11.

$$x=-\frac{5}{2}y-\frac{11}{2}$$

Substitute \frac{-5y-11}{2} for x in the other equation, 2ax+\left(-5b\right)y=41.

$$2a\left(-\frac{5}{2}y-\frac{11}{2}\right)+\left(-5b\right)y=41$$

Multiply 2a times \frac{-5y-11}{2}.

$$\left(-5a\right)y-11a+\left(-5b\right)y=41$$

Add -5ay to -5by.

$$\left(-5a-5b\right)y-11a=41$$

Add 11a to both sides of the equation.

$$\left(-5a-5b\right)y=11a+41$$

Divide both sides by -5a-5b.

$$y=-\frac{11a+41}{5\left(a+b\right)}$$

Substitute -\frac{41+11a}{5\left(a+b\right)} for y in x=-\frac{5}{2}y-\frac{11}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{5}{2}\left(-\frac{11a+41}{5\left(a+b\right)}\right)-\frac{11}{2}$$

Multiply -\frac{5}{2} times -\frac{41+11a}{5\left(a+b\right)}.

$$x=\frac{11a+41}{2\left(a+b\right)}-\frac{11}{2}$$

Add -\frac{11}{2} to \frac{41+11a}{2\left(a+b\right)}.

$$x=\frac{41-11b}{2\left(a+b\right)}$$

The system is now solved.

$$x=\frac{41-11b}{2\left(a+b\right)},y=-\frac{11a+41}{5\left(a+b\right)}$$

Steps Using Matrices

Put the equations in standard form and then use matrices to solve the system of equations.

$$2x+5y=-11,2ax+\left(-5b\right)y=41$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-11\\41\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right))\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right))\left(\begin{matrix}-11\\41\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right))\left(\begin{matrix}-11\\41\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right))\left(\begin{matrix}-11\\41\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5b}{2\left(-5b\right)-5\times 2a}&-\frac{5}{2\left(-5b\right)-5\times 2a}\\-\frac{2a}{2\left(-5b\right)-5\times 2a}&\frac{2}{2\left(-5b\right)-5\times 2a}\end{matrix}\right)\left(\begin{matrix}-11\\41\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{2\left(a+b\right)}&\frac{1}{2\left(a+b\right)}\\\frac{a}{5\left(a+b\right)}&-\frac{1}{5\left(a+b\right)}\end{matrix}\right)\left(\begin{matrix}-11\\41\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{2\left(a+b\right)}\left(-11\right)+\frac{1}{2\left(a+b\right)}\times 41\\\frac{a}{5\left(a+b\right)}\left(-11\right)+\left(-\frac{1}{5\left(a+b\right)}\right)\times 41\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{41-11b}{2\left(a+b\right)}\\-\frac{11a+41}{5\left(a+b\right)}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{41-11b}{2\left(a+b\right)},y=-\frac{11a+41}{5\left(a+b\right)}$$

Steps Using Elimination

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.

$$2x+5y=-11,2ax+\left(-5b\right)y=41$$

To make 2x and 2ax equal, multiply all terms on each side of the first equation by 2a and all terms on each side of the second by 2.

$$2a\times 2x+2a\times 5y=2a\left(-11\right),2\times 2ax+2\left(-5b\right)y=2\times 41$$

Simplify.

$$4ax+10ay=-22a,4ax+\left(-10b\right)y=82$$

Subtract 4ax+\left(-10b\right)y=82 from 4ax+10ay=-22a by subtracting like terms on each side of the equal sign.

$$4ax+\left(-4a\right)x+10ay+10by=-22a-82$$

Add 4ax to -4ax. Terms 4ax and -4ax cancel out, leaving an equation with only one variable that can be solved.

$$10ay+10by=-22a-82$$

Add 10ay to 10by.

$$\left(10a+10b\right)y=-22a-82$$

Divide both sides by 10a+10b.

$$y=-\frac{11a+41}{5\left(a+b\right)}$$

Substitute -\frac{41+11a}{5\left(a+b\right)} for y in 2ax+\left(-5b\right)y=41. Because the resulting equation contains only one variable, you can solve for x directly.

$$2ax+\left(-5b\right)\left(-\frac{11a+41}{5\left(a+b\right)}\right)=41$$

Multiply -5b times -\frac{41+11a}{5\left(a+b\right)}.

$$2ax+\frac{b\left(11a+41\right)}{a+b}=41$$

Subtract \frac{b\left(41+11a\right)}{a+b} from both sides of the equation.

$$2ax=\frac{a\left(41-11b\right)}{a+b}$$

Divide both sides by 2a.

$$x=\frac{41-11b}{2\left(a+b\right)}$$

The system is now solved.

$$x=\frac{41-11b}{2\left(a+b\right)},y=-\frac{11a+41}{5\left(a+b\right)}$$

Solve for x, y

$$\left\{\begin{matrix}x=-\frac{11b-41}{2\left(a+b\right)}\text{, }y=-\frac{11a+41}{5\left(a+b\right)}\text{, }&a\neq -b\\x=\frac{-5y-11}{2}\text{, }y\in \mathrm{R}\text{, }&a=-\frac{41}{11}\text{ and }b=\frac{41}{11}\end{matrix}\right.$$

Steps Using Substitution

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.

$$2x+5y=-11,2ax+\left(-5b\right)y=41$$

Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.

$$2x+5y=-11$$

Subtract 5y from both sides of the equation.

$$2x=-5y-11$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(-5y-11\right)$$

Multiply \frac{1}{2} times -5y-11.

$$x=-\frac{5}{2}y-\frac{11}{2}$$

Substitute \frac{-5y-11}{2} for x in the other equation, 2ax+\left(-5b\right)y=41.

$$2a\left(-\frac{5}{2}y-\frac{11}{2}\right)+\left(-5b\right)y=41$$

Multiply 2a times \frac{-5y-11}{2}.

$$\left(-5a\right)y-11a+\left(-5b\right)y=41$$

Add -5ay to -5by.

$$\left(-5a-5b\right)y-11a=41$$

Add 11a to both sides of the equation.

$$\left(-5a-5b\right)y=11a+41$$

Divide both sides by -5a-5b.

$$y=-\frac{11a+41}{5\left(a+b\right)}$$

Substitute -\frac{41+11a}{5\left(a+b\right)} for y in x=-\frac{5}{2}y-\frac{11}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{5}{2}\left(-\frac{11a+41}{5\left(a+b\right)}\right)-\frac{11}{2}$$

Multiply -\frac{5}{2} times -\frac{41+11a}{5\left(a+b\right)}.

$$x=\frac{11a+41}{2\left(a+b\right)}-\frac{11}{2}$$

Add -\frac{11}{2} to \frac{41+11a}{2\left(a+b\right)}.

$$x=\frac{41-11b}{2\left(a+b\right)}$$

The system is now solved.

$$x=\frac{41-11b}{2\left(a+b\right)},y=-\frac{11a+41}{5\left(a+b\right)}$$

Steps Using Matrices

Put the equations in standard form and then use matrices to solve the system of equations.

$$2x+5y=-11,2ax+\left(-5b\right)y=41$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-11\\41\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right))\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right))\left(\begin{matrix}-11\\41\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right))\left(\begin{matrix}-11\\41\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\2a&-5b\end{matrix}\right))\left(\begin{matrix}-11\\41\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5b}{2\left(-5b\right)-5\times 2a}&-\frac{5}{2\left(-5b\right)-5\times 2a}\\-\frac{2a}{2\left(-5b\right)-5\times 2a}&\frac{2}{2\left(-5b\right)-5\times 2a}\end{matrix}\right)\left(\begin{matrix}-11\\41\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{2\left(a+b\right)}&\frac{1}{2\left(a+b\right)}\\\frac{a}{5\left(a+b\right)}&-\frac{1}{5\left(a+b\right)}\end{matrix}\right)\left(\begin{matrix}-11\\41\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{2\left(a+b\right)}\left(-11\right)+\frac{1}{2\left(a+b\right)}\times 41\\\frac{a}{5\left(a+b\right)}\left(-11\right)+\left(-\frac{1}{5\left(a+b\right)}\right)\times 41\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{41-11b}{2\left(a+b\right)}\\-\frac{11a+41}{5\left(a+b\right)}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{41-11b}{2\left(a+b\right)},y=-\frac{11a+41}{5\left(a+b\right)}$$

Steps Using Elimination

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.

$$2x+5y=-11,2ax+\left(-5b\right)y=41$$

To make 2x and 2ax equal, multiply all terms on each side of the first equation by 2a and all terms on each side of the second by 2.

$$2a\times 2x+2a\times 5y=2a\left(-11\right),2\times 2ax+2\left(-5b\right)y=2\times 41$$

Simplify.

$$4ax+10ay=-22a,4ax+\left(-10b\right)y=82$$

Subtract 4ax+\left(-10b\right)y=82 from 4ax+10ay=-22a by subtracting like terms on each side of the equal sign.

$$4ax+\left(-4a\right)x+10ay+10by=-22a-82$$

Add 4ax to -4ax. Terms 4ax and -4ax cancel out, leaving an equation with only one variable that can be solved.

$$10ay+10by=-22a-82$$

Add 10ay to 10by.

$$\left(10a+10b\right)y=-22a-82$$

Divide both sides by 10a+10b.

$$y=-\frac{11a+41}{5\left(a+b\right)}$$

Substitute -\frac{41+11a}{5\left(a+b\right)} for y in 2ax+\left(-5b\right)y=41. Because the resulting equation contains only one variable, you can solve for x directly.

$$2ax+\left(-5b\right)\left(-\frac{11a+41}{5\left(a+b\right)}\right)=41$$

Multiply -5b times -\frac{41+11a}{5\left(a+b\right)}.

$$2ax+\frac{b\left(11a+41\right)}{a+b}=41$$

Subtract \frac{b\left(41+11a\right)}{a+b} from both sides of the equation.

$$2ax=\frac{a\left(41-11b\right)}{a+b}$$

Divide both sides by 2a.

$$x=\frac{41-11b}{2\left(a+b\right)}$$

The system is now solved.

$$x=\frac{41-11b}{2\left(a+b\right)},y=-\frac{11a+41}{5\left(a+b\right)}$$