Solve {r}{3(x+2)-5y=11}{x-7(y-1)=14}

$$\left.\begin{array} { r } { 3 ( x + 2 ) - 5 y = 11 } \\ { x - 7 ( y - 1 ) = 14 } \end{array} \right\}$$

Solve for x, y

$$x=0$$

$$y=-1$$

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.

$$3\left(x+2\right)-5y=11,x-7\left(y-1\right)=14$$

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

$$3\left(x+2\right)-5y=11$$

Multiply 3 times x+2.

$$3x+6-5y=11$$

Subtract 6 from both sides of the equation.

$$3x-5y=5$$

Add 5y to both sides of the equation.

$$3x=5y+5$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(5y+5\right)$$

Multiply \frac{1}{3} times 5+5y.

$$x=\frac{5}{3}y+\frac{5}{3}$$

Substitute \frac{5+5y}{3} for x in the other equation, x-7\left(y-1\right)=14.

$$\frac{5}{3}y+\frac{5}{3}-7\left(y-1\right)=14$$

Multiply -7 times y-1.

$$\frac{5}{3}y+\frac{5}{3}-7y+7=14$$

Add \frac{5y}{3} to -7y.

$$-\frac{16}{3}y+\frac{5}{3}+7=14$$

Add \frac{5}{3} to 7.

$$-\frac{16}{3}y+\frac{26}{3}=14$$

Subtract \frac{26}{3} from both sides of the equation.

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

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

$$y=-1$$

Substitute -1 for y in x=\frac{5}{3}y+\frac{5}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{5}{3}\left(-1\right)+\frac{5}{3}$$

Multiply \frac{5}{3} times -1.

$$x=\frac{-5+5}{3}$$

Add \frac{5}{3} to -\frac{5}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=0$$

The system is now solved.

$$x=0,y=-1$$

Steps Using Matrices

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

$$3\left(x+2\right)-5y=11,x-7\left(y-1\right)=14$$

Simplify the first equation to put it in standard form.

$$3\left(x+2\right)-5y=11$$

Multiply 3 times x+2.

$$3x+6-5y=11$$

Subtract 6 from both sides of the equation.

$$3x-5y=5$$

Simplify the second equation to put it in standard form.

$$x-7\left(y-1\right)=14$$

Multiply -7 times y-1.

$$x-7y+7=14$$

Subtract 7 from both sides of the equation.

$$x-7y=7$$

Write the equations in matrix form.

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

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-5\\1&-7\end{matrix}\right).

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

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

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

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

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

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.

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

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{16}&-\frac{5}{16}\\\frac{1}{16}&-\frac{3}{16}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{16}\times 5-\frac{5}{16}\times 7\\\frac{1}{16}\times 5-\frac{3}{16}\times 7\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

$$x=0,y=-1$$