Question

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Use matrices to solve the system.
2x + 3y = 4
5x + 8y = 11
Need help solving this

Answer 1

9514 1404 393

Answer:

(x, y) = (-1, 2)

Explanation:

The augmented matrix for this system of equations is ...

`\left[\begin{array}c2&3&4\\5&8&11\end{array}\right]`

Your graphing calculator or web solver will tell you the reduced row-echelon form of this matrix is ...

`\left[\begin{array}c1&0&-1\\0&1&2\end{array}\right]`

The solution is (x, y) = (-1, 2).

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The determinant of the coefficient matrix is ...

(2)(8) -(5)(3) = 1

The inverse of the coefficient matrix is the transpose of the cofactor matrix, divided by the determinant. For a 2×2 matrix, that is the original matrix with the diagonal terms swapped, and the off-diagonal terms negated, then divided by the determinant. Since the determinant is 1, the inverse of the coefficient matrix is ...

`\left[\begin{array}{cc}8&-3\\-5&2\end{array}\right]`

Multiplying this by the constant vector gives the solution vector.

x = (8)(4) +(-3)(11) = 32 -33 = -1

y = (-5)(4) +(2)(11) = -20 +22 = 2

(x, y) = (-1, 2)

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Additional comment

Many modern calculators have so many functions that there are likely many you will never use. It is a good idea to learn to use the functions available for solving the kinds of problems you run across. Recent model calculators make it easy to enter matrices and perform various functions on them.

For solving systems of linear equations, the RREF (reduced row-echelon form) function is pretty useful. The bottom row of the result will tell you the kinds of solutions; the right column(s) will tell you what they are.