Solve {l}{x-5-4(y-3)=26}{3(x-4)+2(y+2)=-9}

$$\left. \begin{array} { l } { x - 5 - 4 ( y - 3 ) = 26 } \\ { 3 ( x - 4 ) + 2 ( y + 2 ) = - 9 } \end{array} \right.$$

Solve for x, y

$$x = \frac{17}{7} = 2\frac{3}{7} \approx 2.428571429$$

$$y = -\frac{29}{7} = -4\frac{1}{7} \approx -4.142857143$$

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.

$$x-4\left(y-3\right)-5=26,3\left(x-4\right)+2\left(y+2\right)=-9$$

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

$$x-4\left(y-3\right)-5=26$$

Multiply -4 times y-3.

$$x-4y+12-5=26$$

Add 12 to -5.

$$x-4y+7=26$$

Subtract 7 from both sides of the equation.

$$x-4y=19$$

Add 4y to both sides of the equation.

$$x=4y+19$$

Substitute 4y+19 for x in the other equation, 3\left(x-4\right)+2\left(y+2\right)=-9.

$$3\left(4y+19-4\right)+2\left(y+2\right)=-9$$

Add 19 to -4.

$$3\left(4y+15\right)+2\left(y+2\right)=-9$$

Multiply 3 times 4y+15.

$$12y+45+2\left(y+2\right)=-9$$

Multiply 2 times y+2.

$$12y+45+2y+4=-9$$

Add 12y to 2y.

$$14y+45+4=-9$$

Add 45 to 4.

$$14y+49=-9$$

Subtract 49 from both sides of the equation.

$$14y=-58$$

Divide both sides by 14.

$$y=-\frac{29}{7}$$

Substitute -\frac{29}{7} for y in x=4y+19. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=4\left(-\frac{29}{7}\right)+19$$

Multiply 4 times -\frac{29}{7}.

$$x=-\frac{116}{7}+19$$

Add 19 to -\frac{116}{7}.

$$x=\frac{17}{7}$$

The system is now solved.

$$x=\frac{17}{7},y=-\frac{29}{7}$$

Steps Using Matrices

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

$$x-4\left(y-3\right)-5=26,3\left(x-4\right)+2\left(y+2\right)=-9$$

Simplify the first equation to put it in standard form.

$$x-4\left(y-3\right)-5=26$$

Multiply -4 times y-3.

$$x-4y+12-5=26$$

Add 12 to -5.

$$x-4y+7=26$$

Subtract 7 from both sides of the equation.

$$x-4y=19$$

Simplify the second equation to put it in standard form.

$$3\left(x-4\right)+2\left(y+2\right)=-9$$

Multiply 3 times x-4.

$$3x-12+2\left(y+2\right)=-9$$

Multiply 2 times y+2.

$$3x-12+2y+4=-9$$

Add -12 to 4.

$$3x+2y-8=-9$$

Add 8 to both sides of the equation.

$$3x+2y=-1$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-4\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}19\\-1\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-4\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-4\\3&2\end{matrix}\right))\left(\begin{matrix}1&-4\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\3&2\end{matrix}\right))\left(\begin{matrix}19\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\3&2\end{matrix}\right))\left(\begin{matrix}19\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\3&2\end{matrix}\right))\left(\begin{matrix}19\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-\left(-4\times 3\right)}&-\frac{-4}{2-\left(-4\times 3\right)}\\-\frac{3}{2-\left(-4\times 3\right)}&\frac{1}{2-\left(-4\times 3\right)}\end{matrix}\right)\left(\begin{matrix}19\\-1\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{2}{7}\\-\frac{3}{14}&\frac{1}{14}\end{matrix}\right)\left(\begin{matrix}19\\-1\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 19+\frac{2}{7}\left(-1\right)\\-\frac{3}{14}\times 19+\frac{1}{14}\left(-1\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{7}\\-\frac{29}{7}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{17}{7},y=-\frac{29}{7}$$