Solve {l}{6.0625*10^-3=(-6.25*10^-4)y+0.0625x}{189.7=13.6*10^3y-13.6*10^3x}

$$\left. \begin{array} { l } { 6.0625 \cdot 10 ^ { - 3 } = ( - 6.25 \cdot 10 ^ { - 4 } ) y + 0.0625 x } \\ { 189.7 = 13.6 \cdot 10 ^ { 3 } y - 13.6 \cdot 10 ^ { 3 } x } \end{array} \right.$$

Solve for y, x

$$x=\frac{1321097}{13464000}\approx 0.098120692$$

$$y=\frac{15089}{134640}\approx 0.112069222$$

Steps Using Substitution

Consider the first equation. Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$6.0625\times \frac{1}{1000}=-6.25\times 10^{-4}y+0.0625x$$

Multiply 6.0625 and \frac{1}{1000} to get \frac{97}{16000}.

$$\frac{97}{16000}=-6.25\times 10^{-4}y+0.0625x$$

Calculate 10 to the power of -4 and get \frac{1}{10000}.

$$\frac{97}{16000}=-6.25\times \frac{1}{10000}y+0.0625x$$

Multiply -6.25 and \frac{1}{10000} to get -\frac{1}{1600}.

$$\frac{97}{16000}=-\frac{1}{1600}y+0.0625x$$

Swap sides so that all variable terms are on the left hand side.

$$-\frac{1}{1600}y+0.0625x=\frac{97}{16000}$$

Consider the second equation. Calculate 10 to the power of 3 and get 1000.

$$189.7=13.6\times 1000y-13.6\times 10^{3}x$$

Multiply 13.6 and 1000 to get 13600.

$$189.7=13600y-13.6\times 10^{3}x$$

Calculate 10 to the power of 3 and get 1000.

$$189.7=13600y-13.6\times 1000x$$

Multiply 13.6 and 1000 to get 13600.

$$189.7=13600y-13600x$$

Swap sides so that all variable terms are on the left hand side.

$$13600y-13600x=189.7$$

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.

$$-\frac{1}{1600}y+0.0625x=\frac{97}{16000},13600y-13600x=189.7$$

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

$$-\frac{1}{1600}y+0.0625x=\frac{97}{16000}$$

Subtract \frac{x}{16} from both sides of the equation.

$$-\frac{1}{1600}y=-0.0625x+\frac{97}{16000}$$

Multiply both sides by -1600.

$$y=-1600\left(-0.0625x+\frac{97}{16000}\right)$$

Multiply -1600 times -\frac{x}{16}+\frac{97}{16000}.

$$y=100x-\frac{97}{10}$$

Substitute 100x-\frac{97}{10} for y in the other equation, 13600y-13600x=189.7.

$$13600\left(100x-\frac{97}{10}\right)-13600x=189.7$$

Multiply 13600 times 100x-\frac{97}{10}.

$$1360000x-131920-13600x=189.7$$

Add 1360000x to -13600x.

$$1346400x-131920=189.7$$

Add 131920 to both sides of the equation.

$$1346400x=132109.7$$

Divide both sides by 1346400.

$$x=\frac{1321097}{13464000}$$

Substitute \frac{1321097}{13464000} for x in y=100x-\frac{97}{10}. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=100\times \frac{1321097}{13464000}-\frac{97}{10}$$

Multiply 100 times \frac{1321097}{13464000}.

$$y=\frac{1321097}{134640}-\frac{97}{10}$$

Add -\frac{97}{10} to \frac{1321097}{134640} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{15089}{134640}$$

The system is now solved.

$$y=\frac{15089}{134640},x=\frac{1321097}{13464000}$$

Steps Using Matrices

Consider the first equation. Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$6.0625\times \frac{1}{1000}=-6.25\times 10^{-4}y+0.0625x$$

Multiply 6.0625 and \frac{1}{1000} to get \frac{97}{16000}.

$$\frac{97}{16000}=-6.25\times 10^{-4}y+0.0625x$$

Calculate 10 to the power of -4 and get \frac{1}{10000}.

$$\frac{97}{16000}=-6.25\times \frac{1}{10000}y+0.0625x$$

Multiply -6.25 and \frac{1}{10000} to get -\frac{1}{1600}.

$$\frac{97}{16000}=-\frac{1}{1600}y+0.0625x$$

Swap sides so that all variable terms are on the left hand side.

$$-\frac{1}{1600}y+0.0625x=\frac{97}{16000}$$

Consider the second equation. Calculate 10 to the power of 3 and get 1000.

$$189.7=13.6\times 1000y-13.6\times 10^{3}x$$

Multiply 13.6 and 1000 to get 13600.

$$189.7=13600y-13.6\times 10^{3}x$$

Calculate 10 to the power of 3 and get 1000.

$$189.7=13600y-13.6\times 1000x$$

Multiply 13.6 and 1000 to get 13600.

$$189.7=13600y-13600x$$

Swap sides so that all variable terms are on the left hand side.

$$13600y-13600x=189.7$$

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

$$-\frac{1}{1600}y+0.0625x=\frac{97}{16000},13600y-13600x=189.7$$

Write the equations in matrix form.

$$\left(\begin{matrix}-\frac{1}{1600}&0.0625\\13600&-13600\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{97}{16000}\\189.7\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-\frac{1}{1600}&0.0625\\13600&-13600\end{matrix}\right).

$$inverse(\left(\begin{matrix}-\frac{1}{1600}&0.0625\\13600&-13600\end{matrix}\right))\left(\begin{matrix}-\frac{1}{1600}&0.0625\\13600&-13600\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-\frac{1}{1600}&0.0625\\13600&-13600\end{matrix}\right))\left(\begin{matrix}\frac{97}{16000}\\189.7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-\frac{1}{1600}&0.0625\\13600&-13600\end{matrix}\right))\left(\begin{matrix}\frac{97}{16000}\\189.7\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-\frac{1}{1600}&0.0625\\13600&-13600\end{matrix}\right))\left(\begin{matrix}\frac{97}{16000}\\189.7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{13600}{-\frac{1}{1600}\left(-13600\right)-0.0625\times 13600}&-\frac{0.0625}{-\frac{1}{1600}\left(-13600\right)-0.0625\times 13600}\\-\frac{13600}{-\frac{1}{1600}\left(-13600\right)-0.0625\times 13600}&-\frac{\frac{1}{1600}}{-\frac{1}{1600}\left(-13600\right)-0.0625\times 13600}\end{matrix}\right)\left(\begin{matrix}\frac{97}{16000}\\189.7\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1600}{99}&\frac{1}{13464}\\\frac{1600}{99}&\frac{1}{1346400}\end{matrix}\right)\left(\begin{matrix}\frac{97}{16000}\\189.7\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1600}{99}\times \frac{97}{16000}+\frac{1}{13464}\times 189.7\\\frac{1600}{99}\times \frac{97}{16000}+\frac{1}{1346400}\times 189.7\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{15089}{134640}\\\frac{1321097}{13464000}\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=\frac{15089}{134640},x=\frac{1321097}{13464000}$$

Steps Using Elimination

Consider the first equation. Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$6.0625\times \frac{1}{1000}=-6.25\times 10^{-4}y+0.0625x$$

Multiply 6.0625 and \frac{1}{1000} to get \frac{97}{16000}.

$$\frac{97}{16000}=-6.25\times 10^{-4}y+0.0625x$$

Calculate 10 to the power of -4 and get \frac{1}{10000}.

$$\frac{97}{16000}=-6.25\times \frac{1}{10000}y+0.0625x$$

Multiply -6.25 and \frac{1}{10000} to get -\frac{1}{1600}.

$$\frac{97}{16000}=-\frac{1}{1600}y+0.0625x$$

Swap sides so that all variable terms are on the left hand side.

$$-\frac{1}{1600}y+0.0625x=\frac{97}{16000}$$

Consider the second equation. Calculate 10 to the power of 3 and get 1000.

$$189.7=13.6\times 1000y-13.6\times 10^{3}x$$

Multiply 13.6 and 1000 to get 13600.

$$189.7=13600y-13.6\times 10^{3}x$$

Calculate 10 to the power of 3 and get 1000.

$$189.7=13600y-13.6\times 1000x$$

Multiply 13.6 and 1000 to get 13600.

$$189.7=13600y-13600x$$

Swap sides so that all variable terms are on the left hand side.

$$13600y-13600x=189.7$$

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.

$$-\frac{1}{1600}y+0.0625x=\frac{97}{16000},13600y-13600x=189.7$$

To make -\frac{y}{1600} and 13600y equal, multiply all terms on each side of the first equation by 13600 and all terms on each side of the second by -\frac{1}{1600}.

$$13600\left(-\frac{1}{1600}\right)y+13600\times 0.0625x=13600\times \frac{97}{16000},-\frac{1}{1600}\times 13600y-\frac{1}{1600}\left(-13600\right)x=-\frac{1}{1600}\times 189.7$$

Simplify.

$$-\frac{17}{2}y+850x=\frac{1649}{20},-\frac{17}{2}y+\frac{17}{2}x=-\frac{1897}{16000}$$

Subtract -\frac{17}{2}y+\frac{17}{2}x=-\frac{1897}{16000} from -\frac{17}{2}y+850x=\frac{1649}{20} by subtracting like terms on each side of the equal sign.

$$-\frac{17}{2}y+\frac{17}{2}y+850x-\frac{17}{2}x=\frac{1649}{20}+\frac{1897}{16000}$$

Add -\frac{17y}{2} to \frac{17y}{2}. Terms -\frac{17y}{2} and \frac{17y}{2} cancel out, leaving an equation with only one variable that can be solved.

$$850x-\frac{17}{2}x=\frac{1649}{20}+\frac{1897}{16000}$$

Add 850x to -\frac{17x}{2}.

$$\frac{1683}{2}x=\frac{1649}{20}+\frac{1897}{16000}$$

Add \frac{1649}{20} to \frac{1897}{16000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$\frac{1683}{2}x=\frac{1321097}{16000}$$

Divide both sides of the equation by \frac{1683}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{1321097}{13464000}$$

Substitute \frac{1321097}{13464000} for x in 13600y-13600x=189.7. Because the resulting equation contains only one variable, you can solve for y directly.

$$13600y-13600\times \frac{1321097}{13464000}=189.7$$

Multiply -13600 times \frac{1321097}{13464000}.

$$13600y-\frac{1321097}{990}=189.7$$

Add \frac{1321097}{990} to both sides of the equation.

$$13600y=\frac{150890}{99}$$

Divide both sides by 13600.

$$y=\frac{15089}{134640}$$

The system is now solved.

$$y=\frac{15089}{134640},x=\frac{1321097}{13464000}$$