Solve {l}{2(x+3)-3y=2}{5x-3(2y-1)=-4}

$$\left\{ \begin{array} { l } { 2 ( x + 3 ) - 3 y = 2 } \\ { 5 x - 3 ( 2 y - 1 ) = - 4 } \end{array} \right\}$$

Solve for x, y

$$x=1$$

$$y=2$$

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.

$$2\left(x+3\right)-3y=2,5x-3\left(2y-1\right)=-4$$

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

$$2\left(x+3\right)-3y=2$$

Multiply 2 times x+3.

$$2x+6-3y=2$$

Subtract 6 from both sides of the equation.

$$2x-3y=-4$$

Add 3y to both sides of the equation.

$$2x=3y-4$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(3y-4\right)$$

Multiply \frac{1}{2} times 3y-4.

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

Substitute \frac{3y}{2}-2 for x in the other equation, 5x-3\left(2y-1\right)=-4.

$$5\left(\frac{3}{2}y-2\right)-3\left(2y-1\right)=-4$$

Multiply 5 times \frac{3y}{2}-2.

$$\frac{15}{2}y-10-3\left(2y-1\right)=-4$$

Multiply -3 times 2y-1.

$$\frac{15}{2}y-10-6y+3=-4$$

Add \frac{15y}{2} to -6y.

$$\frac{3}{2}y-10+3=-4$$

Add -10 to 3.

$$\frac{3}{2}y-7=-4$$

Add 7 to both sides of the equation.

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

Divide both sides of the equation by \frac{3}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=2$$

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

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

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

$$x=3-2$$

Add -2 to 3.

$$x=1$$

The system is now solved.

$$x=1,y=2$$

Steps Using Matrices

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

$$2\left(x+3\right)-3y=2,5x-3\left(2y-1\right)=-4$$

Simplify the first equation to put it in standard form.

$$2\left(x+3\right)-3y=2$$

Multiply 2 times x+3.

$$2x+6-3y=2$$

Subtract 6 from both sides of the equation.

$$2x-3y=-4$$

Simplify the second equation to put it in standard form.

$$5x-3\left(2y-1\right)=-4$$

Multiply -3 times 2y-1.

$$5x-6y+3=-4$$

Subtract 3 from both sides of the equation.

$$5x-6y=-7$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&-3\\5&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\-7\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&-3\\5&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\5&-6\end{matrix}\right))\left(\begin{matrix}2&-3\\5&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&-6\end{matrix}\right))\left(\begin{matrix}-4\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&-6\end{matrix}\right))\left(\begin{matrix}-4\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&-6\end{matrix}\right))\left(\begin{matrix}-4\\-7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{2\left(-6\right)-\left(-3\times 5\right)}&-\frac{-3}{2\left(-6\right)-\left(-3\times 5\right)}\\-\frac{5}{2\left(-6\right)-\left(-3\times 5\right)}&\frac{2}{2\left(-6\right)-\left(-3\times 5\right)}\end{matrix}\right)\left(\begin{matrix}-4\\-7\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2&1\\-\frac{5}{3}&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}-4\\-7\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\left(-4\right)-7\\-\frac{5}{3}\left(-4\right)+\frac{2}{3}\left(-7\right)\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

$$x=1,y=2$$