Solve {l}{0.56x+0.28y=95.2}{0.53(x+y)=106}

$$\left\{ \begin{array} { l } { 0.56 x + 0.28 y = 95.2 } \\ { 0.53 ( x + y ) = 106 } \end{array} \right.$$

Solve for x, y

$$x=140$$

$$y=60$$

Steps Using Substitution

Consider the second equation. Divide both sides by 0.53.

$$x+y=\frac{106}{0.53}$$

Expand \frac{106}{0.53} by multiplying both numerator and the denominator by 100.

$$x+y=\frac{10600}{53}$$

Divide 10600 by 53 to get 200.

$$x+y=200$$

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.

$$0.56x+0.28y=95.2,x+y=200$$

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

$$0.56x+0.28y=95.2$$

Subtract \frac{7y}{25} from both sides of the equation.

$$0.56x=-0.28y+95.2$$

Divide both sides of the equation by 0.56, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{25}{14}\left(-0.28y+95.2\right)$$

Multiply \frac{25}{14} times -\frac{7y}{25}+95.2.

$$x=-0.5y+170$$

Substitute -\frac{y}{2}+170 for x in the other equation, x+y=200.

$$-0.5y+170+y=200$$

Add -\frac{y}{2} to y.

$$0.5y+170=200$$

Subtract 170 from both sides of the equation.

$$0.5y=30$$

Multiply both sides by 2.

$$y=60$$

Substitute 60 for y in x=-0.5y+170. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-0.5\times 60+170$$

Multiply -0.5 times 60.

$$x=-30+170$$

Add 170 to -30.

$$x=140$$

The system is now solved.

$$x=140,y=60$$

Steps Using Matrices

Consider the second equation. Divide both sides by 0.53.

$$x+y=\frac{106}{0.53}$$

Expand \frac{106}{0.53} by multiplying both numerator and the denominator by 100.

$$x+y=\frac{10600}{53}$$

Divide 10600 by 53 to get 200.

$$x+y=200$$

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

$$0.56x+0.28y=95.2,x+y=200$$

Write the equations in matrix form.

$$\left(\begin{matrix}0.56&0.28\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}95.2\\200\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}0.56&0.28\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.56&0.28\\1&1\end{matrix}\right))\left(\begin{matrix}0.56&0.28\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.56&0.28\\1&1\end{matrix}\right))\left(\begin{matrix}95.2\\200\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.56&0.28\\1&1\end{matrix}\right))\left(\begin{matrix}95.2\\200\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.56&0.28\\1&1\end{matrix}\right))\left(\begin{matrix}95.2\\200\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{0.56-0.28}&-\frac{0.28}{0.56-0.28}\\-\frac{1}{0.56-0.28}&\frac{0.56}{0.56-0.28}\end{matrix}\right)\left(\begin{matrix}95.2\\200\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{7}&-1\\-\frac{25}{7}&2\end{matrix}\right)\left(\begin{matrix}95.2\\200\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{7}\times 95.2-200\\-\frac{25}{7}\times 95.2+2\times 200\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}140\\60\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=140,y=60$$

Steps Using Elimination

Consider the second equation. Divide both sides by 0.53.

$$x+y=\frac{106}{0.53}$$

Expand \frac{106}{0.53} by multiplying both numerator and the denominator by 100.

$$x+y=\frac{10600}{53}$$

Divide 10600 by 53 to get 200.

$$x+y=200$$

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.

$$0.56x+0.28y=95.2,x+y=200$$

To make \frac{14x}{25} and x equal, multiply all terms on each side of the first equation by 1 and all terms on each side of the second by 0.56.

$$0.56x+0.28y=95.2,0.56x+0.56y=0.56\times 200$$

Simplify.

$$0.56x+0.28y=95.2,0.56x+0.56y=112$$

Subtract 0.56x+0.56y=112 from 0.56x+0.28y=95.2 by subtracting like terms on each side of the equal sign.

$$0.56x-0.56x+0.28y-0.56y=95.2-112$$

Add \frac{14x}{25} to -\frac{14x}{25}. Terms \frac{14x}{25} and -\frac{14x}{25} cancel out, leaving an equation with only one variable that can be solved.

$$0.28y-0.56y=95.2-112$$

Add \frac{7y}{25} to -\frac{14y}{25}.

$$-0.28y=95.2-112$$

Add 95.2 to -112.

$$-0.28y=-16.8$$

Divide both sides of the equation by -0.28, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=60$$

Substitute 60 for y in x+y=200. Because the resulting equation contains only one variable, you can solve for x directly.

$$x+60=200$$

Subtract 60 from both sides of the equation.

$$x=140$$

The system is now solved.

$$x=140,y=60$$