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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Linear Systems of Equations by Elimination

In some systems it is necessary to multiply each equation by a constant so that one variable will be eliminated when the equations are added.

Example

Use elimination to find the solution of this system.

 3x + 4y = 18 17x + 6y = 52

Solution

Step 1 Eliminate one variable.

Letâ€™s eliminate y.

â€¢ Multiply both sides of the first equation by 3 to make the y-coefficient 12.

3(3x + 4y = 18) 9x + 12y

= 54

â€¢ Multiply both sides of the second equation by -2 to make the y-coefficient -12.  -2(17x + 6y = 52) → -34x - 12y = 104
 9x- 34x +- 12y12y == - 54104 - 25x + 0y = - 50

Simplify. The y-terms have been eliminated.

Divide both sides by -25.

- 25x

x

= - 50

= 2

Step 2 Substitute the value found in Step 1 into either of the original equations and solve.

3x + 4y

= 18

Substitute 2 for x in the first equation.

Multiply.

Subtract 6 from both sides.

Divide both sides by 4.

The solution is (2, 3).

3(2) + 4y

6 + 4y

4y

y

= 18

= 18

= 12

= 3

Step 3 To check the solution, substitute it into each original equation. Then simplify.

Substitute 2 for x and 3 for y into each original equation. Then simplify.

In each case, the result will be a true statement.

The details of the check are left to you.

Note:

 3x + 4y = 18 17x + 6y = 52

In this system, the coefficients of y are 4 and 6, respectively.

Multiples of 4 are 4, 8, 12,...

Multiples of 6 are 6, 12, 18,...

Notice that 12 is the least common multiple of 4 and 6. To make the coefficients of y opposites:

â€¢ multiply both sides of the first equation by 3.

â€¢ multiply both sides of the second equation by -2.