Solve {l}{15y=18x+11y+16}{17x+5y=7y+2}

$$\left\{ \begin{array} { l } { 15 y = 18 x + 11 y + 16 } \\ { 17 x + 5 y = 7 y + 2 } \end{array} \right.$$

Solve for y, x

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

$$y = \frac{77}{8} = 9\frac{5}{8} = 9.625$$

Steps Using Substitution

Consider the first equation. Subtract 18x from both sides.

$$15y-18x=11y+16$$

Subtract 11y from both sides.

$$15y-18x-11y=16$$

Combine 15y and -11y to get 4y.

$$4y-18x=16$$

Consider the second equation. Subtract 7y from both sides.

$$17x+5y-7y=2$$

Combine 5y and -7y to get -2y.

$$17x-2y=2$$

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.

$$4y-18x=16,-2y+17x=2$$

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

$$4y-18x=16$$

Add 18x to both sides of the equation.

$$4y=18x+16$$

Divide both sides by 4.

$$y=\frac{1}{4}\left(18x+16\right)$$

Multiply \frac{1}{4} times 18x+16.

$$y=\frac{9}{2}x+4$$

Substitute \frac{9x}{2}+4 for y in the other equation, -2y+17x=2.

$$-2\left(\frac{9}{2}x+4\right)+17x=2$$

Multiply -2 times \frac{9x}{2}+4.

$$-9x-8+17x=2$$

Add -9x to 17x.

$$8x-8=2$$

Add 8 to both sides of the equation.

$$8x=10$$

Divide both sides by 8.

$$x=\frac{5}{4}$$

Substitute \frac{5}{4} for x in y=\frac{9}{2}x+4. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=\frac{9}{2}\times \frac{5}{4}+4$$

Multiply \frac{9}{2} times \frac{5}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{45}{8}+4$$

Add 4 to \frac{45}{8}.

$$y=\frac{77}{8}$$

The system is now solved.

$$y=\frac{77}{8},x=\frac{5}{4}$$

Steps Using Matrices

Consider the first equation. Subtract 18x from both sides.

$$15y-18x=11y+16$$

Subtract 11y from both sides.

$$15y-18x-11y=16$$

Combine 15y and -11y to get 4y.

$$4y-18x=16$$

Consider the second equation. Subtract 7y from both sides.

$$17x+5y-7y=2$$

Combine 5y and -7y to get -2y.

$$17x-2y=2$$

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

$$4y-18x=16,-2y+17x=2$$

Write the equations in matrix form.

$$\left(\begin{matrix}4&-18\\-2&17\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}16\\2\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&-18\\-2&17\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-18\\-2&17\end{matrix}\right))\left(\begin{matrix}4&-18\\-2&17\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&-18\\-2&17\end{matrix}\right))\left(\begin{matrix}16\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&-18\\-2&17\end{matrix}\right))\left(\begin{matrix}16\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&-18\\-2&17\end{matrix}\right))\left(\begin{matrix}16\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{17}{4\times 17-\left(-18\left(-2\right)\right)}&-\frac{-18}{4\times 17-\left(-18\left(-2\right)\right)}\\-\frac{-2}{4\times 17-\left(-18\left(-2\right)\right)}&\frac{4}{4\times 17-\left(-18\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}16\\2\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{17}{32}&\frac{9}{16}\\\frac{1}{16}&\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}16\\2\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{17}{32}\times 16+\frac{9}{16}\times 2\\\frac{1}{16}\times 16+\frac{1}{8}\times 2\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{77}{8}\\\frac{5}{4}\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=\frac{77}{8},x=\frac{5}{4}$$

Steps Using Elimination

Consider the first equation. Subtract 18x from both sides.

$$15y-18x=11y+16$$

Subtract 11y from both sides.

$$15y-18x-11y=16$$

Combine 15y and -11y to get 4y.

$$4y-18x=16$$

Consider the second equation. Subtract 7y from both sides.

$$17x+5y-7y=2$$

Combine 5y and -7y to get -2y.

$$17x-2y=2$$

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$4y-18x=16,-2y+17x=2$$

To make 4y and -2y equal, multiply all terms on each side of the first equation by -2 and all terms on each side of the second by 4.

$$-2\times 4y-2\left(-18\right)x=-2\times 16,4\left(-2\right)y+4\times 17x=4\times 2$$

Simplify.

$$-8y+36x=-32,-8y+68x=8$$

Subtract -8y+68x=8 from -8y+36x=-32 by subtracting like terms on each side of the equal sign.

$$-8y+8y+36x-68x=-32-8$$

Add -8y to 8y. Terms -8y and 8y cancel out, leaving an equation with only one variable that can be solved.

$$36x-68x=-32-8$$

Add 36x to -68x.

$$-32x=-32-8$$

Add -32 to -8.

$$-32x=-40$$

Divide both sides by -32.

$$x=\frac{5}{4}$$

Substitute \frac{5}{4} for x in -2y+17x=2. Because the resulting equation contains only one variable, you can solve for y directly.

$$-2y+17\times \frac{5}{4}=2$$

Multiply 17 times \frac{5}{4}.

$$-2y+\frac{85}{4}=2$$

Subtract \frac{85}{4} from both sides of the equation.

$$-2y=-\frac{77}{4}$$

Divide both sides by -2.

$$y=\frac{77}{8}$$

The system is now solved.

$$y=\frac{77}{8},x=\frac{5}{4}$$