Solve {l}{x=23y+6}{24x=36y}

$$\left\{ \begin{array} { l } { x = 23 y + 6 } \\ { 24 x = 36 y } \end{array} \right.$$

Solve for x, y

$$x=-\frac{18}{43}\approx -0.418604651$$

$$y=-\frac{12}{43}\approx -0.279069767$$

Steps Using Substitution

Consider the first equation. Subtract 23y from both sides.

$$x-23y=6$$

Consider the second equation. Subtract 36y from both sides.

$$24x-36y=0$$

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.

$$x-23y=6,24x-36y=0$$

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

$$x-23y=6$$

Add 23y to both sides of the equation.

$$x=23y+6$$

Substitute 23y+6 for x in the other equation, 24x-36y=0.

$$24\left(23y+6\right)-36y=0$$

Multiply 24 times 23y+6.

$$552y+144-36y=0$$

Add 552y to -36y.

$$516y+144=0$$

Subtract 144 from both sides of the equation.

$$516y=-144$$

Divide both sides by 516.

$$y=-\frac{12}{43}$$

Substitute -\frac{12}{43} for y in x=23y+6. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=23\left(-\frac{12}{43}\right)+6$$

Multiply 23 times -\frac{12}{43}.

$$x=-\frac{276}{43}+6$$

Add 6 to -\frac{276}{43}.

$$x=-\frac{18}{43}$$

The system is now solved.

$$x=-\frac{18}{43},y=-\frac{12}{43}$$

Steps Using Matrices

Consider the first equation. Subtract 23y from both sides.

$$x-23y=6$$

Consider the second equation. Subtract 36y from both sides.

$$24x-36y=0$$

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

$$x-23y=6,24x-36y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-23\\24&-36\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-23\\24&-36\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-23\\24&-36\end{matrix}\right))\left(\begin{matrix}1&-23\\24&-36\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-23\\24&-36\end{matrix}\right))\left(\begin{matrix}6\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-23\\24&-36\end{matrix}\right))\left(\begin{matrix}6\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-23\\24&-36\end{matrix}\right))\left(\begin{matrix}6\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{36}{-36-\left(-23\times 24\right)}&-\frac{-23}{-36-\left(-23\times 24\right)}\\-\frac{24}{-36-\left(-23\times 24\right)}&\frac{1}{-36-\left(-23\times 24\right)}\end{matrix}\right)\left(\begin{matrix}6\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{43}&\frac{23}{516}\\-\frac{2}{43}&\frac{1}{516}\end{matrix}\right)\left(\begin{matrix}6\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{43}\times 6\\-\frac{2}{43}\times 6\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{18}{43}\\-\frac{12}{43}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{18}{43},y=-\frac{12}{43}$$

Steps Using Elimination

Consider the first equation. Subtract 23y from both sides.

$$x-23y=6$$

Consider the second equation. Subtract 36y from both sides.

$$24x-36y=0$$

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.

$$x-23y=6,24x-36y=0$$

To make x and 24x equal, multiply all terms on each side of the first equation by 24 and all terms on each side of the second by 1.

$$24x+24\left(-23\right)y=24\times 6,24x-36y=0$$

Simplify.

$$24x-552y=144,24x-36y=0$$

Subtract 24x-36y=0 from 24x-552y=144 by subtracting like terms on each side of the equal sign.

$$24x-24x-552y+36y=144$$

Add 24x to -24x. Terms 24x and -24x cancel out, leaving an equation with only one variable that can be solved.

$$-552y+36y=144$$

Add -552y to 36y.

$$-516y=144$$

Divide both sides by -516.

$$y=-\frac{12}{43}$$

Substitute -\frac{12}{43} for y in 24x-36y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$24x-36\left(-\frac{12}{43}\right)=0$$

Multiply -36 times -\frac{12}{43}.

$$24x+\frac{432}{43}=0$$

Subtract \frac{432}{43} from both sides of the equation.

$$24x=-\frac{432}{43}$$

Divide both sides by 24.

$$x=-\frac{18}{43}$$

The system is now solved.

$$x=-\frac{18}{43},y=-\frac{12}{43}$$