Solve {l}{2(2x)=6y}{4x+2(6y)=360}

$$\left. \begin{array} { l } { 2 ( 2 x ) = 6 y } \\ { 4 x + 2 ( 6 y ) = 360 } \end{array} \right.$$

Solve for x, y

$$x=30$$

$$y=20$$

Steps Using Substitution

Consider the first equation. Multiply 2 and 2 to get 4.

$$4x=6y$$

Divide both sides by 4.

$$x=\frac{1}{4}\times 6y$$

Multiply \frac{1}{4} times 6y.

$$x=\frac{3}{2}y$$

Substitute \frac{3y}{2} for x in the other equation, 4x+12y=360.

$$4\times \frac{3}{2}y+12y=360$$

Multiply 4 times \frac{3y}{2}.

$$6y+12y=360$$

Add 6y to 12y.

$$18y=360$$

Divide both sides by 18.

$$y=20$$

Substitute 20 for y in x=\frac{3}{2}y. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{3}{2}\times 20$$

Multiply \frac{3}{2} times 20.

$$x=30$$

The system is now solved.

$$x=30,y=20$$

Steps Using Matrices

Consider the first equation. Multiply 2 and 2 to get 4.

$$4x=6y$$

Subtract 6y from both sides.

$$4x-6y=0$$

Consider the second equation. Multiply 2 and 6 to get 12.

$$4x+12y=360$$

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

$$4x-6y=0,4x+12y=360$$

Write the equations in matrix form.

$$\left(\begin{matrix}4&-6\\4&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\360\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&-6\\4&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-6\\4&12\end{matrix}\right))\left(\begin{matrix}4&-6\\4&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-6\\4&12\end{matrix}\right))\left(\begin{matrix}0\\360\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-6\\4&12\end{matrix}\right))\left(\begin{matrix}0\\360\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-6\\4&12\end{matrix}\right))\left(\begin{matrix}0\\360\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{4\times 12-\left(-6\times 4\right)}&-\frac{-6}{4\times 12-\left(-6\times 4\right)}\\-\frac{4}{4\times 12-\left(-6\times 4\right)}&\frac{4}{4\times 12-\left(-6\times 4\right)}\end{matrix}\right)\left(\begin{matrix}0\\360\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{12}\\-\frac{1}{18}&\frac{1}{18}\end{matrix}\right)\left(\begin{matrix}0\\360\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}\times 360\\\frac{1}{18}\times 360\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\20\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=30,y=20$$

Steps Using Elimination

Consider the first equation. Multiply 2 and 2 to get 4.

$$4x=6y$$

Subtract 6y from both sides.

$$4x-6y=0$$

Consider the second equation. Multiply 2 and 6 to get 12.

$$4x+12y=360$$

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.

$$4x-6y=0,4x+12y=360$$

Subtract 4x+12y=360 from 4x-6y=0 by subtracting like terms on each side of the equal sign.

$$4x-4x-6y-12y=-360$$

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

$$-6y-12y=-360$$

Add -6y to -12y.

$$-18y=-360$$

Divide both sides by -18.

$$y=20$$

Substitute 20 for y in 4x+12y=360. Because the resulting equation contains only one variable, you can solve for x directly.

$$4x+12\times 20=360$$

Multiply 12 times 20.

$$4x+240=360$$

Subtract 240 from both sides of the equation.

$$4x=120$$

Divide both sides by 4.

$$x=30$$

The system is now solved.

$$x=30,y=20$$