Solve {l}{xcos(15)+ysin(10)=433}{xsin(15)=ycos(10)}

$$\left. \begin{array} { l } { x \cos ( 15 ) + y \sin ( 10 ) = 433 } \\ { x \sin ( 15 ) = y \cos ( 10 ) } \end{array} \right.$$

Solve for x_0, y_0

$$x_{0} = \frac{2132108785271430320000000000000000000}{4980973490458727396200334333832221} = 428\frac{2.521313550949717 \times 10^{32}}{4.9809734904587275 \times 10^{33}} \approx 428.050618891$$

$$y_{0} = \frac{560343232646957402100000000000000000}{4980973490458727396200334333832221} = 112\frac{2.4742017155799743 \times 10^{33}}{4.9809734904587275 \times 10^{33}} \approx 112.496730553$$

Steps Using Substitution

Evaluate trigonometric functions in the problem

$$\left. \begin{array} { l } { x 0.9659258262890683 + y 0.17364817766693033 = 433 }\\ { x 0.25881904510252074 = y 0.984807753012208 } \end{array} \right.$$

Consider the second equation. Subtract y_{0}\times 0.984807753012208 from both sides.

$$x_{0}\times 0.25881904510252074-y_{0}\times 0.984807753012208=0$$

Multiply -1 and 0.984807753012208 to get -0.984807753012208.

$$x_{0}\times 0.25881904510252074-0.984807753012208y_{0}=0$$

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.

$$0.9659258262890683x_{0}+0.17364817766693033y_{0}=433,0.25881904510252074x_{0}-0.984807753012208y_{0}=0$$

Choose one of the equations and solve it for x_{0} by isolating x_{0} on the left hand side of the equal sign.

$$0.9659258262890683x_{0}+0.17364817766693033y_{0}=433$$

Subtract \frac{17364817766693033y_{0}}{100000000000000000} from both sides of the equation.

$$0.9659258262890683x_{0}=-0.17364817766693033y_{0}+433$$

Divide both sides of the equation by 0.9659258262890683, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x_{0}=\frac{10000000000000000}{9659258262890683}\left(-0.17364817766693033y_{0}+433\right)$$

Multiply \frac{10000000000000000}{9659258262890683} times -\frac{17364817766693033y_{0}}{100000000000000000}+433.

$$x_{0}=-\frac{17364817766693033}{96592582628906830}y_{0}+\frac{4330000000000000000}{9659258262890683}$$

Substitute -\frac{17364817766693033y_{0}}{96592582628906830}+\frac{4330000000000000000}{9659258262890683} for x_{0} in the other equation, 0.25881904510252074x_{0}-0.984807753012208y_{0}=0.

$$0.25881904510252074\left(-\frac{17364817766693033}{96592582628906830}y_{0}+\frac{4330000000000000000}{9659258262890683}\right)-0.984807753012208y_{0}=0$$

Multiply 0.25881904510252074 times -\frac{17364817766693033y_{0}}{96592582628906830}+\frac{4330000000000000000}{9659258262890683}.

$$-\frac{224717277637738878831000104800221}{4829629131445341500000000000000000}y_{0}+\frac{5603432326469574021}{48296291314453415}-0.984807753012208y_{0}=0$$

Add -\frac{224717277637738878831000104800221y_{0}}{4829629131445341500000000000000000} to -\frac{61550484563263y_{0}}{62500000000000}.

$$-\frac{4980973490458727396200334333832221}{4829629131445341500000000000000000}y_{0}+\frac{5603432326469574021}{48296291314453415}=0$$

Subtract \frac{5603432326469574021}{48296291314453415} from both sides of the equation.

$$-\frac{4980973490458727396200334333832221}{4829629131445341500000000000000000}y_{0}=-\frac{5603432326469574021}{48296291314453415}$$

Divide both sides of the equation by -\frac{4980973490458727396200334333832221}{4829629131445341500000000000000000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y_{0}=\frac{560343232646957402100000000000000000}{4980973490458727396200334333832221}$$

Substitute \frac{560343232646957402100000000000000000}{4980973490458727396200334333832221} for y_{0} in x_{0}=-\frac{17364817766693033}{96592582628906830}y_{0}+\frac{4330000000000000000}{9659258262890683}. Because the resulting equation contains only one variable, you can solve for x_{0} directly.

$$x_{0}=-\frac{17364817766693033}{96592582628906830}\times \frac{560343232646957402100000000000000000}{4980973490458727396200334333832221}+\frac{4330000000000000000}{9659258262890683}$$

Multiply -\frac{17364817766693033}{96592582628906830} times \frac{560343232646957402100000000000000000}{4980973490458727396200334333832221} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x_{0}=-\frac{973025812171409345338230453784956930000000000000000}{48112509344952909183156077514661049201520586096943}+\frac{4330000000000000000}{9659258262890683}$$

Add \frac{4330000000000000000}{9659258262890683} to -\frac{973025812171409345338230453784956930000000000000000}{48112509344952909183156077514661049201520586096943} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x_{0}=\frac{2132108785271430320000000000000000000}{4980973490458727396200334333832221}$$

The system is now solved.

$$x_{0}=\frac{2132108785271430320000000000000000000}{4980973490458727396200334333832221},y_{0}=\frac{560343232646957402100000000000000000}{4980973490458727396200334333832221}$$

Steps Using Matrices

Evaluate trigonometric functions in the problem

$$\left. \begin{array} { l } { x 0.9659258262890683 + y 0.17364817766693033 = 433 }\\ { x 0.25881904510252074 = y 0.984807753012208 } \end{array} \right.$$

Consider the second equation. Subtract y_{0}\times 0.984807753012208 from both sides.

$$x_{0}\times 0.25881904510252074-y_{0}\times 0.984807753012208=0$$

Multiply -1 and 0.984807753012208 to get -0.984807753012208.

$$x_{0}\times 0.25881904510252074-0.984807753012208y_{0}=0$$

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

$$0.9659258262890683x_{0}+0.17364817766693033y_{0}=433,0.25881904510252074x_{0}-0.984807753012208y_{0}=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}0.9659258262890683&0.17364817766693033\\0.25881904510252074&-0.984807753012208\end{matrix}\right)\left(\begin{matrix}x_{0}\\y_{0}\end{matrix}\right)=\left(\begin{matrix}433\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}0.9659258262890683&0.17364817766693033\\0.25881904510252074&-0.984807753012208\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.9659258262890683&0.17364817766693033\\0.25881904510252074&-0.984807753012208\end{matrix}\right))\left(\begin{matrix}0.9659258262890683&0.17364817766693033\\0.25881904510252074&-0.984807753012208\end{matrix}\right)\left(\begin{matrix}x_{0}\\y_{0}\end{matrix}\right)=inverse(\left(\begin{matrix}0.9659258262890683&0.17364817766693033\\0.25881904510252074&-0.984807753012208\end{matrix}\right))\left(\begin{matrix}433\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x_{0}\\y_{0}\end{matrix}\right)=inverse(\left(\begin{matrix}0.9659258262890683&0.17364817766693033\\0.25881904510252074&-0.984807753012208\end{matrix}\right))\left(\begin{matrix}433\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x_{0}\\y_{0}\end{matrix}\right)=inverse(\left(\begin{matrix}0.9659258262890683&0.17364817766693033\\0.25881904510252074&-0.984807753012208\end{matrix}\right))\left(\begin{matrix}433\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x_{0}\\y_{0}\end{matrix}\right)=\left(\begin{matrix}-\frac{0.984807753012208}{0.9659258262890683\left(-0.984807753012208\right)-0.17364817766693033\times 0.25881904510252074}&-\frac{0.17364817766693033}{0.9659258262890683\left(-0.984807753012208\right)-0.17364817766693033\times 0.25881904510252074}\\-\frac{0.25881904510252074}{0.9659258262890683\left(-0.984807753012208\right)-0.17364817766693033\times 0.25881904510252074}&\frac{0.9659258262890683}{0.9659258262890683\left(-0.984807753012208\right)-0.17364817766693033\times 0.25881904510252074}\end{matrix}\right)\left(\begin{matrix}433\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x_{0}\\y_{0}\end{matrix}\right)=\left(\begin{matrix}\frac{4924038765061040000000000000000000}{4980973490458727396200334333832221}&\frac{868240888334651650000000000000000}{4980973490458727396200334333832221}\\\frac{1294095225512603700000000000000000}{4980973490458727396200334333832221}&-\frac{4829629131445341500000000000000000}{4980973490458727396200334333832221}\end{matrix}\right)\left(\begin{matrix}433\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x_{0}\\y_{0}\end{matrix}\right)=\left(\begin{matrix}\frac{4924038765061040000000000000000000}{4980973490458727396200334333832221}\times 433\\\frac{1294095225512603700000000000000000}{4980973490458727396200334333832221}\times 433\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x_{0}\\y_{0}\end{matrix}\right)=\left(\begin{matrix}\frac{2132108785271430320000000000000000000}{4980973490458727396200334333832221}\\\frac{560343232646957402100000000000000000}{4980973490458727396200334333832221}\end{matrix}\right)$$

Extract the matrix elements x_{0} and y_{0}.

$$x_{0}=\frac{2132108785271430320000000000000000000}{4980973490458727396200334333832221},y_{0}=\frac{560343232646957402100000000000000000}{4980973490458727396200334333832221}$$

Steps Using Elimination

Evaluate trigonometric functions in the problem

$$\left. \begin{array} { l } { x 0.9659258262890683 + y 0.17364817766693033 = 433 }\\ { x 0.25881904510252074 = y 0.984807753012208 } \end{array} \right.$$

Consider the second equation. Subtract y_{0}\times 0.984807753012208 from both sides.

$$x_{0}\times 0.25881904510252074-y_{0}\times 0.984807753012208=0$$

Multiply -1 and 0.984807753012208 to get -0.984807753012208.

$$x_{0}\times 0.25881904510252074-0.984807753012208y_{0}=0$$

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.

$$0.9659258262890683x_{0}+0.17364817766693033y_{0}=433,0.25881904510252074x_{0}-0.984807753012208y_{0}=0$$

To make \frac{9659258262890683x_{0}}{10000000000000000} and \frac{12940952255126037x_{0}}{50000000000000000} equal, multiply all terms on each side of the first equation by 0.25881904510252074 and all terms on each side of the second by 0.9659258262890683.

$$0.25881904510252074\times 0.9659258262890683x_{0}+0.25881904510252074\times 0.17364817766693033y_{0}=0.25881904510252074\times 433,0.9659258262890683\times 0.25881904510252074x_{0}+0.9659258262890683\left(-0.984807753012208\right)y_{0}=0$$

Simplify.

$$0.249999999999999981842040236026542x_{0}+0.0449434555275477757662000209600442y_{0}=112.06864652939148042,0.249999999999999981842040236026542x_{0}-0.9512512425641977034738668458064y_{0}=0$$

Subtract 0.249999999999999981842040236026542x_{0}-0.9512512425641977034738668458064y_{0}=0 from 0.249999999999999981842040236026542x_{0}+0.0449434555275477757662000209600442y_{0}=112.06864652939148042 by subtracting like terms on each side of the equal sign.

$$0.249999999999999981842040236026542x_{0}-0.249999999999999981842040236026542x_{0}+0.0449434555275477757662000209600442y_{0}+0.9512512425641977034738668458064y_{0}=112.06864652939148042$$

Add \frac{124999999999999990921020118013271x_{0}}{500000000000000000000000000000000} to -\frac{124999999999999990921020118013271x_{0}}{500000000000000000000000000000000}. Terms \frac{124999999999999990921020118013271x_{0}}{500000000000000000000000000000000} and -\frac{124999999999999990921020118013271x_{0}}{500000000000000000000000000000000} cancel out, leaving an equation with only one variable that can be solved.

$$0.0449434555275477757662000209600442y_{0}+0.9512512425641977034738668458064y_{0}=112.06864652939148042$$

Add \frac{224717277637738878831000104800221y_{0}}{5000000000000000000000000000000000} to \frac{594532026602623564671166778629y_{0}}{625000000000000000000000000000}.

$$0.9961946980917454792400668667664442y_{0}=112.06864652939148042$$

Divide both sides of the equation by 0.9961946980917454792400668667664442, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y_{0}=\frac{560343232646957402100000000000000000}{4980973490458727396200334333832221}$$

Substitute \frac{560343232646957402100000000000000000}{4980973490458727396200334333832221} for y_{0} in 0.25881904510252074x_{0}-0.984807753012208y_{0}=0. Because the resulting equation contains only one variable, you can solve for x_{0} directly.

$$0.25881904510252074x_{0}-0.984807753012208\times \frac{560343232646957402100000000000000000}{4980973490458727396200334333832221}=0$$

Multiply -0.984807753012208 times \frac{560343232646957402100000000000000000}{4980973490458727396200334333832221} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$0.25881904510252074x_{0}-\frac{551830359858647031632972537264836800}{4980973490458727396200334333832221}=0$$

Add \frac{551830359858647031632972537264836800}{4980973490458727396200334333832221} to both sides of the equation.

$$0.25881904510252074x_{0}=\frac{551830359858647031632972537264836800}{4980973490458727396200334333832221}$$

Divide both sides of the equation by 0.25881904510252074, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x_{0}=\frac{2132108785271430320000000000000000000}{4980973490458727396200334333832221}$$

The system is now solved.

$$x_{0}=\frac{2132108785271430320000000000000000000}{4980973490458727396200334333832221},y_{0}=\frac{560343232646957402100000000000000000}{4980973490458727396200334333832221}$$