Solve {l}{x+7y=27}{347x+245y=3} | Microsoft Math Solver (2024)

\left\{ \begin{array} { l } { x + 7 y = 27 } \\ { 347 x + 245 y = 3 } \end{array} \right.

Solve for x, y

x = -\frac{157}{52} = -3\frac{1}{52} \approx -3.019230769

y = \frac{223}{52} = 4\frac{15}{52} \approx 4.288461538

Solve {l}{x+7y=27}{347x+245y=3} | Microsoft Math Solver (1)

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x+7y=27,347x+245y=3

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.

x+7y=27

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

x=-7y+27

Subtract 7y from both sides of the equation.

347\left(-7y+27\right)+245y=3

Substitute -7y+27 for x in the other equation, 347x+245y=3.

-2429y+9369+245y=3

Multiply 347 times -7y+27.

-2184y+9369=3

Add -2429y to 245y.

-2184y=-9366

Subtract 9369 from both sides of the equation.

y=\frac{223}{52}

Divide both sides by -2184.

x=-7\times \frac{223}{52}+27

Substitute \frac{223}{52} for y in x=-7y+27. Because the resulting equation contains only one variable, you can solve for x directly.

x=-\frac{1561}{52}+27

Multiply -7 times \frac{223}{52}.

x=-\frac{157}{52}

Add 27 to -\frac{1561}{52}.

x=-\frac{157}{52},y=\frac{223}{52}

The system is now solved.

x+7y=27,347x+245y=3

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

\left(\begin{matrix}1&7\\347&245\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}27\\3\end{matrix}\right)

Write the equations in matrix form.

inverse(\left(\begin{matrix}1&7\\347&245\end{matrix}\right))\left(\begin{matrix}1&7\\347&245\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\347&245\end{matrix}\right))\left(\begin{matrix}27\\3\end{matrix}\right)

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&7\\347&245\end{matrix}\right).

\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\347&245\end{matrix}\right))\left(\begin{matrix}27\\3\end{matrix}\right)

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

\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\347&245\end{matrix}\right))\left(\begin{matrix}27\\3\end{matrix}\right)

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

\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{245}{245-7\times 347}&-\frac{7}{245-7\times 347}\\-\frac{347}{245-7\times 347}&\frac{1}{245-7\times 347}\end{matrix}\right)\left(\begin{matrix}27\\3\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{35}{312}&\frac{1}{312}\\\frac{347}{2184}&-\frac{1}{2184}\end{matrix}\right)\left(\begin{matrix}27\\3\end{matrix}\right)

Do the arithmetic.

\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{35}{312}\times 27+\frac{1}{312}\times 3\\\frac{347}{2184}\times 27-\frac{1}{2184}\times 3\end{matrix}\right)

Multiply the matrices.

\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{157}{52}\\\frac{223}{52}\end{matrix}\right)

Do the arithmetic.

x=-\frac{157}{52},y=\frac{223}{52}

Extract the matrix elements x and y.

x+7y=27,347x+245y=3

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.

347x+347\times 7y=347\times 27,347x+245y=3

To make x and 347x equal, multiply all terms on each side of the first equation by 347 and all terms on each side of the second by 1.

347x+2429y=9369,347x+245y=3

Simplify.

347x-347x+2429y-245y=9369-3

Subtract 347x+245y=3 from 347x+2429y=9369 by subtracting like terms on each side of the equal sign.

2429y-245y=9369-3

Add 347x to -347x. Terms 347x and -347x cancel out, leaving an equation with only one variable that can be solved.

2184y=9369-3

Add 2429y to -245y.

2184y=9366

Add 9369 to -3.

y=\frac{223}{52}

Divide both sides by 2184.

347x+245\times \frac{223}{52}=3

Substitute \frac{223}{52} for y in 347x+245y=3. Because the resulting equation contains only one variable, you can solve for x directly.

347x+\frac{54635}{52}=3

Multiply 245 times \frac{223}{52}.

347x=-\frac{54479}{52}

Subtract \frac{54635}{52} from both sides of the equation.

x=-\frac{157}{52}

Divide both sides by 347.

x=-\frac{157}{52},y=\frac{223}{52}

The system is now solved.

Solve {l}{x+7y=27}{347x+245y=3} | Microsoft Math Solver (2024)

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