Solve {l}{20x+9y=20}{-4x-2y=-42}

$$\left. \begin{array} { l } { 20 x + 9 y = 20 } \\ { - 4 x - 2 y = - 42 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{169}{2} = -84\frac{1}{2} = -84.5$$

$$y=190$$

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.

$$20x+9y=20,-4x-2y=-42$$

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

$$20x+9y=20$$

Subtract 9y from both sides of the equation.

$$20x=-9y+20$$

Divide both sides by 20.

$$x=\frac{1}{20}\left(-9y+20\right)$$

Multiply \frac{1}{20} times -9y+20.

$$x=-\frac{9}{20}y+1$$

Substitute -\frac{9y}{20}+1 for x in the other equation, -4x-2y=-42.

$$-4\left(-\frac{9}{20}y+1\right)-2y=-42$$

Multiply -4 times -\frac{9y}{20}+1.

$$\frac{9}{5}y-4-2y=-42$$

Add \frac{9y}{5} to -2y.

$$-\frac{1}{5}y-4=-42$$

Add 4 to both sides of the equation.

$$-\frac{1}{5}y=-38$$

Multiply both sides by -5.

$$y=190$$

Substitute 190 for y in x=-\frac{9}{20}y+1. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{9}{20}\times 190+1$$

Multiply -\frac{9}{20} times 190.

$$x=-\frac{171}{2}+1$$

Add 1 to -\frac{171}{2}.

$$x=-\frac{169}{2}$$

The system is now solved.

$$x=-\frac{169}{2},y=190$$

Steps Using Matrices

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

$$20x+9y=20,-4x-2y=-42$$

Write the equations in matrix form.

$$\left(\begin{matrix}20&9\\-4&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\-42\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}20&9\\-4&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}20&9\\-4&-2\end{matrix}\right))\left(\begin{matrix}20&9\\-4&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}20&9\\-4&-2\end{matrix}\right))\left(\begin{matrix}20\\-42\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}20&9\\-4&-2\end{matrix}\right))\left(\begin{matrix}20\\-42\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}20&9\\-4&-2\end{matrix}\right))\left(\begin{matrix}20\\-42\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{20\left(-2\right)-9\left(-4\right)}&-\frac{9}{20\left(-2\right)-9\left(-4\right)}\\-\frac{-4}{20\left(-2\right)-9\left(-4\right)}&\frac{20}{20\left(-2\right)-9\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}20\\-42\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{9}{4}\\-1&-5\end{matrix}\right)\left(\begin{matrix}20\\-42\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 20+\frac{9}{4}\left(-42\right)\\-20-5\left(-42\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{169}{2}\\190\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{169}{2},y=190$$

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.

$$20x+9y=20,-4x-2y=-42$$

To make 20x and -4x equal, multiply all terms on each side of the first equation by -4 and all terms on each side of the second by 20.

$$-4\times 20x-4\times 9y=-4\times 20,20\left(-4\right)x+20\left(-2\right)y=20\left(-42\right)$$

Simplify.

$$-80x-36y=-80,-80x-40y=-840$$

Subtract -80x-40y=-840 from -80x-36y=-80 by subtracting like terms on each side of the equal sign.

$$-80x+80x-36y+40y=-80+840$$

Add -80x to 80x. Terms -80x and 80x cancel out, leaving an equation with only one variable that can be solved.

$$-36y+40y=-80+840$$

Add -36y to 40y.

$$4y=-80+840$$

Add -80 to 840.

$$4y=760$$

Divide both sides by 4.

$$y=190$$

Substitute 190 for y in -4x-2y=-42. Because the resulting equation contains only one variable, you can solve for x directly.

$$-4x-2\times 190=-42$$

Multiply -2 times 190.

$$-4x-380=-42$$

Add 380 to both sides of the equation.

$$-4x=338$$

Divide both sides by -4.

$$x=-\frac{169}{2}$$

The system is now solved.

$$x=-\frac{169}{2},y=190$$