Solve {l}{26x+40y-27=-20x+5y+100}{40x+50y-16=5x+4y+100}

$$\left\{ \begin{array} { l } { 26 x + 40 y - 27 = - 20 x + 5 y + 100 } \\ { 40 x + 50 y - 16 = 5 x + 4 y + 100 } \end{array} \right.$$

Solve for x, y

$$x=2$$

$$y=1$$

Steps Using Substitution

Consider the first equation. Add 20x to both sides.

$$26x+40y-27+20x=5y+100$$

Combine 26x and 20x to get 46x.

$$46x+40y-27=5y+100$$

Subtract 5y from both sides.

$$46x+40y-27-5y=100$$

Combine 40y and -5y to get 35y.

$$46x+35y-27=100$$

Add 27 to both sides.

$$46x+35y=100+27$$

Add 100 and 27 to get 127.

$$46x+35y=127$$

Consider the second equation. Subtract 5x from both sides.

$$40x+50y-16-5x=4y+100$$

Combine 40x and -5x to get 35x.

$$35x+50y-16=4y+100$$

Subtract 4y from both sides.

$$35x+50y-16-4y=100$$

Combine 50y and -4y to get 46y.

$$35x+46y-16=100$$

Add 16 to both sides.

$$35x+46y=100+16$$

Add 100 and 16 to get 116.

$$35x+46y=116$$

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.

$$46x+35y=127,35x+46y=116$$

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

$$46x+35y=127$$

Subtract 35y from both sides of the equation.

$$46x=-35y+127$$

Divide both sides by 46.

$$x=\frac{1}{46}\left(-35y+127\right)$$

Multiply \frac{1}{46} times -35y+127.

$$x=-\frac{35}{46}y+\frac{127}{46}$$

Substitute \frac{-35y+127}{46} for x in the other equation, 35x+46y=116.

$$35\left(-\frac{35}{46}y+\frac{127}{46}\right)+46y=116$$

Multiply 35 times \frac{-35y+127}{46}.

$$-\frac{1225}{46}y+\frac{4445}{46}+46y=116$$

Add -\frac{1225y}{46} to 46y.

$$\frac{891}{46}y+\frac{4445}{46}=116$$

Subtract \frac{4445}{46} from both sides of the equation.

$$\frac{891}{46}y=\frac{891}{46}$$

Divide both sides of the equation by \frac{891}{46}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=1$$

Substitute 1 for y in x=-\frac{35}{46}y+\frac{127}{46}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{-35+127}{46}$$

Add \frac{127}{46} to -\frac{35}{46} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=2$$

The system is now solved.

$$x=2,y=1$$

Steps Using Matrices

Consider the first equation. Add 20x to both sides.

$$26x+40y-27+20x=5y+100$$

Combine 26x and 20x to get 46x.

$$46x+40y-27=5y+100$$

Subtract 5y from both sides.

$$46x+40y-27-5y=100$$

Combine 40y and -5y to get 35y.

$$46x+35y-27=100$$

Add 27 to both sides.

$$46x+35y=100+27$$

Add 100 and 27 to get 127.

$$46x+35y=127$$

Consider the second equation. Subtract 5x from both sides.

$$40x+50y-16-5x=4y+100$$

Combine 40x and -5x to get 35x.

$$35x+50y-16=4y+100$$

Subtract 4y from both sides.

$$35x+50y-16-4y=100$$

Combine 50y and -4y to get 46y.

$$35x+46y-16=100$$

Add 16 to both sides.

$$35x+46y=100+16$$

Add 100 and 16 to get 116.

$$35x+46y=116$$

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

$$46x+35y=127,35x+46y=116$$

Write the equations in matrix form.

$$\left(\begin{matrix}46&35\\35&46\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}127\\116\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}46&35\\35&46\end{matrix}\right).

$$inverse(\left(\begin{matrix}46&35\\35&46\end{matrix}\right))\left(\begin{matrix}46&35\\35&46\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}46&35\\35&46\end{matrix}\right))\left(\begin{matrix}127\\116\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}46&35\\35&46\end{matrix}\right))\left(\begin{matrix}127\\116\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}46&35\\35&46\end{matrix}\right))\left(\begin{matrix}127\\116\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{46}{46\times 46-35\times 35}&-\frac{35}{46\times 46-35\times 35}\\-\frac{35}{46\times 46-35\times 35}&\frac{46}{46\times 46-35\times 35}\end{matrix}\right)\left(\begin{matrix}127\\116\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{46}{891}&-\frac{35}{891}\\-\frac{35}{891}&\frac{46}{891}\end{matrix}\right)\left(\begin{matrix}127\\116\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{46}{891}\times 127-\frac{35}{891}\times 116\\-\frac{35}{891}\times 127+\frac{46}{891}\times 116\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\1\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=2,y=1$$

Steps Using Elimination

Consider the first equation. Add 20x to both sides.

$$26x+40y-27+20x=5y+100$$

Combine 26x and 20x to get 46x.

$$46x+40y-27=5y+100$$

Subtract 5y from both sides.

$$46x+40y-27-5y=100$$

Combine 40y and -5y to get 35y.

$$46x+35y-27=100$$

Add 27 to both sides.

$$46x+35y=100+27$$

Add 100 and 27 to get 127.

$$46x+35y=127$$

Consider the second equation. Subtract 5x from both sides.

$$40x+50y-16-5x=4y+100$$

Combine 40x and -5x to get 35x.

$$35x+50y-16=4y+100$$

Subtract 4y from both sides.

$$35x+50y-16-4y=100$$

Combine 50y and -4y to get 46y.

$$35x+46y-16=100$$

Add 16 to both sides.

$$35x+46y=100+16$$

Add 100 and 16 to get 116.

$$35x+46y=116$$

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.

$$46x+35y=127,35x+46y=116$$

To make 46x and 35x equal, multiply all terms on each side of the first equation by 35 and all terms on each side of the second by 46.

$$35\times 46x+35\times 35y=35\times 127,46\times 35x+46\times 46y=46\times 116$$

Simplify.

$$1610x+1225y=4445,1610x+2116y=5336$$

Subtract 1610x+2116y=5336 from 1610x+1225y=4445 by subtracting like terms on each side of the equal sign.

$$1610x-1610x+1225y-2116y=4445-5336$$

Add 1610x to -1610x. Terms 1610x and -1610x cancel out, leaving an equation with only one variable that can be solved.

$$1225y-2116y=4445-5336$$

Add 1225y to -2116y.

$$-891y=4445-5336$$

Add 4445 to -5336.

$$-891y=-891$$

Divide both sides by -891.

$$y=1$$

Substitute 1 for y in 35x+46y=116. Because the resulting equation contains only one variable, you can solve for x directly.

$$35x+46=116$$

Subtract 46 from both sides of the equation.

$$35x=70$$

Divide both sides by 35.

$$x=2$$

The system is now solved.

$$x=2,y=1$$