Solve {l}{3(x+2)-y=17}{7x+3(y-1)=12}

$$\left. \begin{array} { l } { 3 ( x + 2 ) - y = 17 } \\ { 7 x + 3 ( y - 1 ) = 12 } \end{array} \right.$$

Solve for x, y

$$x=3$$

$$y=-2$$

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.

$$3\left(x+2\right)-y=17,7x+3\left(y-1\right)=12$$

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

$$3\left(x+2\right)-y=17$$

Multiply 3 times x+2.

$$3x+6-y=17$$

Subtract 6 from both sides of the equation.

$$3x-y=11$$

Add y to both sides of the equation.

$$3x=y+11$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(y+11\right)$$

Multiply \frac{1}{3} times y+11.

$$x=\frac{1}{3}y+\frac{11}{3}$$

Substitute \frac{11+y}{3} for x in the other equation, 7x+3\left(y-1\right)=12.

$$7\left(\frac{1}{3}y+\frac{11}{3}\right)+3\left(y-1\right)=12$$

Multiply 7 times \frac{11+y}{3}.

$$\frac{7}{3}y+\frac{77}{3}+3\left(y-1\right)=12$$

Multiply 3 times y-1.

$$\frac{7}{3}y+\frac{77}{3}+3y-3=12$$

Add \frac{7y}{3} to 3y.

$$\frac{16}{3}y+\frac{77}{3}-3=12$$

Add \frac{77}{3} to -3.

$$\frac{16}{3}y+\frac{68}{3}=12$$

Subtract \frac{68}{3} from both sides of the equation.

$$\frac{16}{3}y=-\frac{32}{3}$$

Divide both sides of the equation by \frac{16}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-2$$

Substitute -2 for y in x=\frac{1}{3}y+\frac{11}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{1}{3}\left(-2\right)+\frac{11}{3}$$

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

$$x=\frac{-2+11}{3}$$

Add \frac{11}{3} to -\frac{2}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=3$$

The system is now solved.

$$x=3,y=-2$$

Steps Using Matrices

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

$$3\left(x+2\right)-y=17,7x+3\left(y-1\right)=12$$

Simplify the first equation to put it in standard form.

$$3\left(x+2\right)-y=17$$

Multiply 3 times x+2.

$$3x+6-y=17$$

Subtract 6 from both sides of the equation.

$$3x-y=11$$

Simplify the second equation to put it in standard form.

$$7x+3\left(y-1\right)=12$$

Multiply 3 times y-1.

$$7x+3y-3=12$$

Add 3 to both sides of the equation.

$$7x+3y=15$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-1\\7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\15\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-1\\7&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-1\\7&3\end{matrix}\right))\left(\begin{matrix}3&-1\\7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\7&3\end{matrix}\right))\left(\begin{matrix}11\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\7&3\end{matrix}\right))\left(\begin{matrix}11\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\7&3\end{matrix}\right))\left(\begin{matrix}11\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-\left(-7\right)}&-\frac{-1}{3\times 3-\left(-7\right)}\\-\frac{7}{3\times 3-\left(-7\right)}&\frac{3}{3\times 3-\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}11\\15\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{16}&\frac{1}{16}\\-\frac{7}{16}&\frac{3}{16}\end{matrix}\right)\left(\begin{matrix}11\\15\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{16}\times 11+\frac{1}{16}\times 15\\-\frac{7}{16}\times 11+\frac{3}{16}\times 15\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

$$x=3,y=-2$$