FreeAlgebra Tutorials!

Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

The addition method can be used to eliminate any variable whose coefficients are opposites. If neither variable has coefficients that are opposites, then we use the multiplication property of equality to change the coefficients of the variables, as shown in Examples 1 and 2.

Example 1

Solve the system by the addition method:

2x - 3y = -13

5x - 12y = -46

Solution

If we multiply both sides of the first equation by -4, the coefficients of y will be 12 and -12, and y will be eliminated by addition.

 (-4)(2x - 3y) = (-4)(-13) Multiply each side by -4 5x - 12y = -46 -8x + 12y = 52 5x - 12y = -46 Add -3x = 6 x = -2

Replace x by -2 in one of the original equations to find y:

 2x - 3y = -13 2(-2) - 3y = -13 -4 - 3y = -13 -3y = -9 y = 3

Because 2(-2) - 3(3) = -13 and 5(-2) - 12(3) = -46 are both true, the solution set is {(-2, 3)}.

Example 2

Solve the system by the addition method:

-2x + 3y = 6

3x - 5y = -11

Solution

To eliminate x, we multiply the first equation by 3 and the second by 2:

 3(-2x + 3y) = 3(6) Multiply each side by 3. 2(3x - 5y) = 2(-11) Multiply each side by 2. -6x + 9y = 18 6x - 10y = -22 Add. -y = -4 y = 4

Note that we could have eliminated y by multiplying by 5 and 3. Now insert y = 4 into one of the original equations to find x:

 -2x + 3(4) = 6 Let y = 4 in -2x + 3y = 6. -2x + 12 = 6 -2x = -6 x = 3

Check that (3, 4) satisfies both equations. The solution set is {(3, 4)}.

We can always use the addition method as long as the equations in a system are in the same form.