Solve {l}{4(x+2)+3(y+1)=15}{5(x+1)+4y=11}

$$\left\{ \begin{array} { l } { 4 ( x + 2 ) + 3 ( y + 1 ) = 15 } \\ { 5 ( x + 1 ) + 4 y = 11 } \end{array} \right.$$

Solve for x, y

$$x=-2$$

$$y=4$$

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.

$$4\left(x+2\right)+3\left(y+1\right)=15,5\left(x+1\right)+4y=11$$

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

$$4\left(x+2\right)+3\left(y+1\right)=15$$

Multiply 4 times x+2.

$$4x+8+3\left(y+1\right)=15$$

Multiply 3 times y+1.

$$4x+8+3y+3=15$$

Add 8 to 3.

$$4x+3y+11=15$$

Subtract 11 from both sides of the equation.

$$4x+3y=4$$

Subtract 3y from both sides of the equation.

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

Divide both sides by 4.

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

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

$$x=-\frac{3}{4}y+1$$

Substitute -\frac{3y}{4}+1 for x in the other equation, 5\left(x+1\right)+4y=11.

$$5\left(-\frac{3}{4}y+1+1\right)+4y=11$$

Add 1 to 1.

$$5\left(-\frac{3}{4}y+2\right)+4y=11$$

Multiply 5 times -\frac{3y}{4}+2.

$$-\frac{15}{4}y+10+4y=11$$

Add -\frac{15y}{4} to 4y.

$$\frac{1}{4}y+10=11$$

Subtract 10 from both sides of the equation.

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

Multiply both sides by 4.

$$y=4$$

Substitute 4 for y in x=-\frac{3}{4}y+1. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{4}\times 4+1$$

Multiply -\frac{3}{4} times 4.

$$x=-3+1$$

Add 1 to -3.

$$x=-2$$

The system is now solved.

$$x=-2,y=4$$

Steps Using Matrices

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

$$4\left(x+2\right)+3\left(y+1\right)=15,5\left(x+1\right)+4y=11$$

Simplify the first equation to put it in standard form.

$$4\left(x+2\right)+3\left(y+1\right)=15$$

Multiply 4 times x+2.

$$4x+8+3\left(y+1\right)=15$$

Multiply 3 times y+1.

$$4x+8+3y+3=15$$

Add 8 to 3.

$$4x+3y+11=15$$

Subtract 11 from both sides of the equation.

$$4x+3y=4$$

Simplify the second equation to put it in standard form.

$$5\left(x+1\right)+4y=11$$

Multiply 5 times x+1.

$$5x+5+4y=11$$

Subtract 5 from both sides of the equation.

$$5x+4y=6$$

Write the equations in matrix form.

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

Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&3\\5&4\end{matrix}\right).

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

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

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

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

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

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.

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

Do the arithmetic.

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

Multiply the matrices.

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

Do the arithmetic.

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

Extract the matrix elements x and y.

$$x=-2,y=4$$