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

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

Solve for x, y

$$x=-20$$

$$y=-10$$

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.

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

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

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

Multiply -5 times y+1.

$$2x-5y-5=5$$

Add 5 to both sides of the equation.

$$2x-5y=10$$

Add 5y to both sides of the equation.

$$2x=5y+10$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(5y+10\right)$$

Multiply \frac{1}{2} times 10+5y.

$$x=\frac{5}{2}y+5$$

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

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

Multiply -1 times 5+\frac{5y}{2}.

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

Multiply 2 times y+2.

$$-\frac{5}{2}y-5+2y+4=4$$

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

$$-\frac{1}{2}y-5+4=4$$

Add -5 to 4.

$$-\frac{1}{2}y-1=4$$

Add 1 to both sides of the equation.

$$-\frac{1}{2}y=5$$

Multiply both sides by -2.

$$y=-10$$

Substitute -10 for y in x=\frac{5}{2}y+5. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{5}{2}\left(-10\right)+5$$

Multiply \frac{5}{2} times -10.

$$x=-25+5$$

Add 5 to -25.

$$x=-20$$

The system is now solved.

$$x=-20,y=-10$$

Steps Using Matrices

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

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

Simplify the first equation to put it in standard form.

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

Multiply -5 times y+1.

$$2x-5y-5=5$$

Add 5 to both sides of the equation.

$$2x-5y=10$$

Simplify the second equation to put it in standard form.

$$-x+2\left(y+2\right)=4$$

Multiply 2 times y+2.

$$-x+2y+4=4$$

Subtract 4 from both sides of the equation.

$$-x+2y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&-5\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&-5\\-1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-5\\-1&2\end{matrix}\right))\left(\begin{matrix}2&-5\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\-1&2\end{matrix}\right))\left(\begin{matrix}10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\-1&2\end{matrix}\right))\left(\begin{matrix}10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\-1&2\end{matrix}\right))\left(\begin{matrix}10\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-\left(-5\left(-1\right)\right)}&-\frac{-5}{2\times 2-\left(-5\left(-1\right)\right)}\\-\frac{-1}{2\times 2-\left(-5\left(-1\right)\right)}&\frac{2}{2\times 2-\left(-5\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}10\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2&-5\\-1&-2\end{matrix}\right)\left(\begin{matrix}10\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\times 10\\-10\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-20\\-10\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-20,y=-10$$