Solve {l}{60(x+0.1)+30(y+0.2)=60}{90x+60y=81}

$$\left\{ \begin{array} { l } { 60 ( x + 0.1 ) + 30 ( y + 0.2 ) = 60 } \\ { 90 x + 60 y = 81 } \end{array} \right.$$

Solve for x, y

$$x=0.5$$

$$y=0.6$$

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.

$$60\left(x+0.1\right)+30\left(y+0.2\right)=60,90x+60y=81$$

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

$$60\left(x+0.1\right)+30\left(y+0.2\right)=60$$

Multiply 60 times x+0.1.

$$60x+6+30\left(y+0.2\right)=60$$

Multiply 30 times y+0.2.

$$60x+6+30y+6=60$$

Add 6 to 6.

$$60x+30y+12=60$$

Subtract 12 from both sides of the equation.

$$60x+30y=48$$

Subtract 30y from both sides of the equation.

$$60x=-30y+48$$

Divide both sides by 60.

$$x=\frac{1}{60}\left(-30y+48\right)$$

Multiply \frac{1}{60} times -30y+48.

$$x=-\frac{1}{2}y+\frac{4}{5}$$

Substitute -\frac{y}{2}+\frac{4}{5} for x in the other equation, 90x+60y=81.

$$90\left(-\frac{1}{2}y+\frac{4}{5}\right)+60y=81$$

Multiply 90 times -\frac{y}{2}+\frac{4}{5}.

$$-45y+72+60y=81$$

Add -45y to 60y.

$$15y+72=81$$

Subtract 72 from both sides of the equation.

$$15y=9$$

Divide both sides by 15.

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

Substitute \frac{3}{5} for y in x=-\frac{1}{2}y+\frac{4}{5}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1}{2}\times \frac{3}{5}+\frac{4}{5}$$

Multiply -\frac{1}{2} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{3}{10}+\frac{4}{5}$$

Add \frac{4}{5} to -\frac{3}{10} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{1}{2}$$

The system is now solved.

$$x=\frac{1}{2},y=\frac{3}{5}$$

Steps Using Matrices

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

$$60\left(x+0.1\right)+30\left(y+0.2\right)=60,90x+60y=81$$

Simplify the first equation to put it in standard form.

$$60\left(x+0.1\right)+30\left(y+0.2\right)=60$$

Multiply 60 times x+0.1.

$$60x+6+30\left(y+0.2\right)=60$$

Multiply 30 times y+0.2.

$$60x+6+30y+6=60$$

Add 6 to 6.

$$60x+30y+12=60$$

Subtract 12 from both sides of the equation.

$$60x+30y=48$$

Write the equations in matrix form.

$$\left(\begin{matrix}60&30\\90&60\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}48\\81\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}60&30\\90&60\end{matrix}\right).

$$inverse(\left(\begin{matrix}60&30\\90&60\end{matrix}\right))\left(\begin{matrix}60&30\\90&60\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&30\\90&60\end{matrix}\right))\left(\begin{matrix}48\\81\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&30\\90&60\end{matrix}\right))\left(\begin{matrix}48\\81\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&30\\90&60\end{matrix}\right))\left(\begin{matrix}48\\81\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{60}{60\times 60-30\times 90}&-\frac{30}{60\times 60-30\times 90}\\-\frac{90}{60\times 60-30\times 90}&\frac{60}{60\times 60-30\times 90}\end{matrix}\right)\left(\begin{matrix}48\\81\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}&-\frac{1}{30}\\-\frac{1}{10}&\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}48\\81\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}\times 48-\frac{1}{30}\times 81\\-\frac{1}{10}\times 48+\frac{1}{15}\times 81\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\\\frac{3}{5}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{1}{2},y=\frac{3}{5}$$