Solve {l}{0.04x+0.02y=5}{0.5(x-2)-0.4y=29}

$$\left. \begin{array} { l } { 0.04 x + 0.02 y = 5 } \\ { 0.5 ( x - 2 ) - 0.4 y = 29 } \end{array} \right.$$

Solve for x, y

$$x=100$$

$$y=50$$

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.

$$0.04x+0.02y=5,0.5\left(x-2\right)-0.4y=29$$

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

$$0.04x+0.02y=5$$

Subtract \frac{y}{50} from both sides of the equation.

$$0.04x=-0.02y+5$$

Multiply both sides by 25.

$$x=25\left(-0.02y+5\right)$$

Multiply 25 times -\frac{y}{50}+5.

$$x=-0.5y+125$$

Substitute -\frac{y}{2}+125 for x in the other equation, 0.5\left(x-2\right)-0.4y=29.

$$0.5\left(-0.5y+125-2\right)-0.4y=29$$

Add 125 to -2.

$$0.5\left(-0.5y+123\right)-0.4y=29$$

Multiply 0.5 times -\frac{y}{2}+123.

$$-0.25y+61.5-0.4y=29$$

Add -\frac{y}{4} to -\frac{2y}{5}.

$$-0.65y+61.5=29$$

Subtract 61.5 from both sides of the equation.

$$-0.65y=-32.5$$

Divide both sides of the equation by -0.65, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=50$$

Substitute 50 for y in x=-0.5y+125. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-0.5\times 50+125$$

Multiply -0.5 times 50.

$$x=-25+125$$

Add 125 to -25.

$$x=100$$

The system is now solved.

$$x=100,y=50$$

Steps Using Matrices

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

$$0.04x+0.02y=5,0.5\left(x-2\right)-0.4y=29$$

Simplify the second equation to put it in standard form.

$$0.5\left(x-2\right)-0.4y=29$$

Multiply 0.5 times x-2.

$$0.5x-1-0.4y=29$$

Add 1 to both sides of the equation.

$$0.5x-0.4y=30$$

Write the equations in matrix form.

$$\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\30\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right))\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{0.4}{0.04\left(-0.4\right)-0.02\times 0.5}&-\frac{0.02}{0.04\left(-0.4\right)-0.02\times 0.5}\\-\frac{0.5}{0.04\left(-0.4\right)-0.02\times 0.5}&\frac{0.04}{0.04\left(-0.4\right)-0.02\times 0.5}\end{matrix}\right)\left(\begin{matrix}5\\30\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{200}{13}&\frac{10}{13}\\\frac{250}{13}&-\frac{20}{13}\end{matrix}\right)\left(\begin{matrix}5\\30\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{200}{13}\times 5+\frac{10}{13}\times 30\\\frac{250}{13}\times 5-\frac{20}{13}\times 30\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}100\\50\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=100,y=50$$