Simultaneous equations: Section 2 Example 4. Solve this system of simultaneous equations:
Solution. If we add the equations as they are, neither one of the unknowns will cancel. Now, if the coefficient of y in equation 2) were −4, then the y's would cancel. Therefore we will expand our strategy as follows: Make one pair of coefficients negatives of one another -- by multiplying To make the coefficients of the y's 4 and −4, we will multiply both sides of equation 2) by 4 :
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 y, substitute x = 5 in either one of the original equations. In equation 1):
The solution is (5, 1). The student should always verify the solution by replacing x and y with (5, 1) in the original equations. Example 5. Solve simultaneously:
Solution. We must make one pair of coefficients negatives of one another. In this example, we must decide which of the unknowns to eliminate, x or y. In either case, we will make the new coefficients the Lowest Common Multiple (LCM) of the original coefficients -- but with opposite signs. Thus, if we eliminate x, then we will make the new coeffients 6 and −6. (The LCM of 3 and 2 is 6.) While if we eliminate y, we will make their new coefficients 10 and −10. (The LCM of 2 and 5 is 10.) Let us choose to eliminate x:
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, they will cancel. To solve for x, we will substitute y = −1 in the original equation 1):
The solution is (0, −1). Problem 3. Solve simultaneously.
To make the y's cancel, multiply equation 2) by 3:
To solve for y: Substitute x = 2 in one of the original equations.
The solution is (2, 3). Problem 4. Solve simultaneously.
To make the x's cancel, multiply equation 1) by −2:
To solve for x: Substitute y = −1 in one of the original equations.
The solution is (1, −1). We could have eliminated y by multiplying equation 1) by 3 and equation 2) by 2. Problem 5. Solve simultaneously:
To make the y's cancel: Multiply equation 1) by 3 and equation 2) by 4:
To solve for y: Substitute x = 3 in one of the original equations.
The solution is (3, 2). Problem 6. Solve simultaneously:
To make the x's cancel: Multiply equation 1) by 2 and equation 2) by −3:
To solve for x: Substitute y = 1 in one of the original equations.
The solution is (−2, 1). We could have eliminated y by multiplying equation 1) by 5 and equation 2) by −2. Problem 7. Solve simultaneously:
To make the x's cancel: Multiply equation 1) by 2 and equation 2) by −5:
To solve for x: Substitute y = −2 in one of the original equations.
The solution is (−1, −2). We could have eliminated y by multiplying equation 1) by 4 and equation 2) by −3. Cramer's Rule A system of two equations in two unknowns has this form: The a's are the coefficients of the x's. The b's are the coefficients of the y's. The following is the matrix of those coefficients. The number a1b2 − b1a2 is called the determinant of that matrix.
Let us denote that determinant by D. Now consider this matrix in which the c's replace the coefficients of the x's: Then the determinant of that matrix -- which we will call Dx -- is c1b2 − b1c2 And consider this matrix in which the c's replace the coeffients of the y's: The determinant of that matrix -- Dy -- is a1c2 − c1a2 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):
Solution.
Therefore,
Problem. Use Cramer's Rule to solve these simultaneous equations.
Therefore,
Section 3: Three equations in three unknowns Please make a donation to keep TheMathPage online. Copyright © 2001-2007 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |