Part Three – Solving equations by Elimination

Holt Algebra I

Text pages 330-334

Solve using substitution after manipulating equations in standard

form.

2x + 3y = 21

-3x – 3y = -12

• Which value, x or y, should we work with first?

• This looks like a very long, drawn-out problem. Is there a better way?

Let’s solve by elimination.

• This method uses opposites to eliminate one of the variables.

• Which variable should be eliminated?

2x + 3y = 21

-3x – 3y = -12

2x + 3y = 21-3x – 3y = -12

Notice that the coefficients with the y value are opposites. (+3 and -3).

Use Columns

Solve for remaining variable.

Substitute that value.

If we combine these two equations together in columns, we can eliminate the y values.

We will solve for x and then insert it’s value into one of the original equations to solve for y.

The steps and explanations

2x + 3y = 21-3x – 3y = -12-1x + 0 = 9

-1x + 0= 9

-1 -1

x = -9

• Add terms from top to bottom.+2x - 3x

+3y - 3y

• Divide both sides by -1.

• Now go back and insert -9 for x.

2x + 3y = 21-3x – 3y = -12

You may insert (x= -9) into either one.

• 2(-9) + 3y = 21• -18 + 3y = 21(add 18 to both sides)

• +3y = 39

3 3

y = 13

• Solution (-9, 13)

• -3(-9) – 3y = -12• +27 – 3y = -12(subtract 27 from both sides)

-3y = -39

-3 -3

y = 13

Try One.

-4x + 3y = -1

4x + 6y = 5

Eliminate the x values.

-4x + 3y = -1

4x + 6y = 5

9y = 4

9y = 4

9 9

y = 4/9

• Solve for x.

• 4x + 6(4/9) = 5

• 4x +24/9 = 45/9

• Subtract 21/9 from both sides.

• 4x = 2 1/3

• Go to the next slide…

4x =21/9

• Divide both sided by 4.

• 4x =21/9

• 4 4

• x =

• x =

To divide fractions, multiply by the reciprocal

21 4

9 1

21 1

9 4x

21 7

36 12

Ready to go one more step?

• What if you don’t have an easy choice.

• You may find that neither equation has opposite coefficients.

11x + 2y = -8

8x + 3y = 5

Let’s try 11x + 2y = -8 and 8x + 3y = 5• Goal

• Observe

• Multiply

• eliminate a variable using opposite coefficients.

• It looks like we should use 2y and 3y since they are smaller numbers

• both sides of the top equation by -3 and both sides of the bottom by 2, we should get coefficients of 6 and -6.

Multiply both sides

(11x + 2y) = (-8) (8x + 3y) = (5)

• -3(11x + 2y) = (-8)-3

• 2(8x + 3y) = (5)2

• We’ll put all four values into parentheses.

• Multiply both sides of the top by -3

• Multiply both sides of the second equation by 2.

Results of the First Steps• -3(11x + 2y) = (-8)-3• 2(8x + 3y) = (5)2

-----------------------• -33x – 6y = +24• 16x + 6y = +10

• -17x + 0 = 34

• From the previous slide

• Use the distributive property

• Now eliminate

From previous slide -17x = 34 x = -2

• 11x + 2y = -8

• 11(-2) + 2y = -8

• -22+ 2y = -8

• 2y = 14

• y = 7

• Pick one of the original equations.

• Solve for the other variable.

• Add 22 to both sides. -8 +22 = 14.

• Solution (-2, 7)

One more for practice

• 3x - 2y = 2• 4x – 7y = 33

--------------------

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

3(4x – 7y) = (33)3

-------------------------

Solution on the next slide…

One more for practice - Solution

• 3x - 2y = 2• 4x – 7y = 33

--------------------• -4(3x - 2y) = (2)-4• 3(4x – 7y) = (33)3

-------------------------

-12x + 8y = -8

12x – 21y = 99

-----------------------

-13y = 91

-13y = 91

-13 -13

y= -7

---------------------------

3x-2(-7)= 2

3x + 14 = 2

3x = -12

x= -4

-------------

Solution (-4, -7)

Which way of solving works best for you?

Graphing?

Substitution?

Elimination?

Make sure you know them all in order to pick the best way to solve each problem.

Assignment: 334:15-33 & 41-47 odds