Lesson 35 Simultaneous equations: Section 2 Cramer's Rule: The method of determinants Example 4. Solve this system of simultaneous equations:
Make one pair of coefficients To make the coefficients of the
The 4 over the arrow in equation 2) signifies that both sides of that equation have been multiplied by 4. Equation 1) has not been changed. To solve for
The solution is (5, 1). The student should always verify the solution by replacing Example 5. Solve simultaneously:
Thus, if we eliminate Let us choose to eliminate
Equation 1) has been multiplied by 2. Equation 2) has been multiplied by −3 -- because we want to make those coefficients 6 and −6, so that on adding, the To solve for
The solution is (0, −1). Problem 3. Solve simultaneously.
To make the
To solve for Substitute
The solution is (2, 3). Problem 4. Solve simultaneously.
To make the
To solve for Substitute
The solution is (1, −1). We could have eliminated Problem 5. Solve simultaneously:
To make the Multiply equation 1) by 3 and equation 2) by 4:
To solve for Substitute
The solution is (3, 2). Problem 6. Solve simultaneously:
To make the Multiply equation 1) by 2 and equation 2) by −3:
To solve for Substitute
The solution is (−2, 1). We could have eliminated Problem 7. Solve simultaneously:
To make the Multiply equation 1) by 2 and equation 2) by −5:
To solve for Substitute
The solution is (−1, −2). We could have eliminated Cramer's Rule: The method of determinants A system of two equations in two unknowns has this form: The The
Let us denote that determinant by D. Now consider this matrix in which the Then the determinant of that matrix -- which we will call D
And consider this matrix in which the The determinant of that matrix -- D
Cramer's Rule then states the following: In every system of two equations in two unknowns
Example. Use Cramer's Rule to solve this system of equations (Problem 7):
Therefore,
Problem. Use Cramer's Rule to solve these simultaneous equations.
Therefore,
When the determinant D is not 0, we say that the equations are When the determinant D is 0, then either 1) there is not a unique solution, it is possible to name many; or 2) there is no solution at all. In case 1), the equations are
2 In case 2), the equations are
Section 3: Three equations in three unknowns Next Lesson: Word problems that lead to simultaneous equations Copyright © 2018 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com Private tutoring available. |