Question

Solve
by using
crammer's rule
3x-y=6
2x-15=-3y​

Answer 1

Answer:

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

both sides of an equation by the same number.  Upon adding the equations, that unknown will be eliminated.

To make the coefficients of the y's 4 and −4, we will multiply both sides of equation 2) by 4 :

1)    3x  +  4y  =  19  simultaneous equations  3x  +  4y  =  19

2)    2x  −  y  =  9  simultaneous equations  8x  −  4y  =  36

  simultaneous equations

  11x        =  55

        x  =  55

11

        x  =  5

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):

3· 5 + 4y  =  19

4y  =  19 − 15

4y  =  4

y  =  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:

1)    3x  +  2y  =  −2

2)    2x  +  5y  =  −5

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:

1)    3x  +  2y  =  −2  simultaneous equations  6x  +  4y  =  −4

2)    2x  +  5y  =  −5  simultaneous equations  −6x  −  15y  =  15

  ________________________________________________________________________

     −  11y  =  11

        y  =  −1.

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):

3x + 2(−1)  =  −2

3x − 2  =  −2

3x  =  0

x  =  0

The solution is (0, −1).

Step-by-step explanation: