Practice Makes Perfect Algebra - Carolyn Wheater [39]
4(2x + 3y) = (19)·4
3(3x − 4y) = (3)·3
becomes
8x + 12y = 76
9x − 12y = 9
Then the equations are added. 8 12 76
Plugging the value of x back in produces the value of y.
2·5 + 3y = 19 ⇒ 3y = 9 ⇒ y = 3
To understand the origins of Cramer’s rule, we want to focus for a moment on two lines, one from each of the solutions above: and . The denominator of 17, common to both, is equal to 2 · 4 + 3 · 3, the opposite of the determinant of the coefficient matrix. . The 85 can be produced from a determinant involving the coefficients of the y-terms and the constants, and the 51 from the coefficients of the x-terms and the constants. Cramer’s rule recognizes this and uses determinants to arrive at the solution of the system quickly and easily.
The denominator
To use Cramer’s rule to solve a system, we first find the determinant of the matrix of coefficients, placing the x coefficients in the first column and the y coefficients in the second: . There is no need to worry about the sign, as you will see in a moment. This determinant will be our denominator for both x and y.
A numerator for each variable
Next we create a determinant for each variable. The numerator for x is the determinant formed when we take the coefficient matrix and replace the values in the x column with the constants.
The numerator for y is the determinant formed when we take the coefficient matrix and replace the values in the y column with the constants.
Notice that both of these have come out opposite in sign to the values we saw in the algebraic solutions. In a moment, when we divide, these sign differences will cancel one another.
Solutions
To find the values of x and y, all that remains is to divide and simplify. The value of x is the x numerator over the denominator: . The value of y is the y numerator over the denominator:
If we put all this work in symbolic terms, we can say that the solution of the system can be expressed as
* * *
EXERCISE 13.7
Solve each system by Cramer’s rule.
1. −3x + 2y = −20
5x − 3y = 33
2. 2x − 7y = 1
3x + 5y = 17
3. 2x + 3y = 2.9
4x − y = −3.3
4. 2x − y − 3z = 10
x − 2y + 3z = −22
3x + 5y − z = 63
5. x + y − z = −6
x − y + z = 14
x − y − z = 8
6. 2x + 3y = 5
2y + z = 5
x + 3z = −5
* * *
Method of inverses
We have already seen that Cramer’s rule allows us to solve systems of equations by means of the determinants of matrices formed from the coefficients of the system. With the addition of inverses to the matrix tool kit, we can expand our understanding of matrix algebra to access two additional methods.
From your experience solving systems, you no doubt recognize that there is no difference in the solution of the system and the system . You know that the particular letters used as variables are less significant than the coefficients and constants that define the system. All the matrix methods of solving systems depend on matrices made from these coefficients and constants, but unlike Cramer’s rule, these methods do not replace the coefficients with the constants.
Matrices of coefficients and of constants
The first of the methods we will explore organizes the critical numbers into two matrices, one for the coefficients of the variables and one for the constants. If we use the system from our earlier discussion as our example, we can form these two matrices. The first is a 2 × 2 matrix containing the coefficients of x in the first column and the coefficients of y in the second column. Each equation forms a row of the matrix, so the matrix of coefficients is . The second matrix, a column matrix with dimension 2 × 1, contains the constants from the other side of each equation. The matrix of constants thus formed is .
The key to this method lies in the fact that the system can be represented by a single statement of matrix multiplication. If the coefficient matrix is multiplied by a column matrix containing our variables, the result is exactly the left side of our system.
Since this is the case, our system of equations is equivalent to the