1. 5.3 Solving Systems of Linear Equations by the Addition Method

2. Solving Using Addition Method
Write both eqns. in standard form (Ax + By = C).
Get opposite coefficients for one of the variables. You may need to mult. one or both eqns. by a nonzero number to do this.
Add the eqns., vertically.
Solve the remaining eqn.
Substitute the value for the variable from step 4 into one of the original eqns. and solve for the other variable.
Check soln. in BOTH eqns., if necessary.

3. Ex. Solve by the addition method: x + y = 3 x – y = 5
Done
x + y = 3
x – y = 5
3. x + y = 3
4. 2x + 0 = 8
2x = 8
x = 4
5. x + y = y = 3 sub 4 for x y + 4 – 4 = 3 – 4 y = -1 Soln: {(4, -1)} 6. Check: x + y = 3 x – y = (-1) = 3 4 – (-1) = 3 3 = = 5 5 = 5
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4. Ex. Solve by the addition method: x + y = 9 -x + y = -3
Done
2. x + y = 9
-x + y = -3
3. x + y = 9
y = 6
2y = 6
y = 3
5. x + y = 9 x + 3 = 9 sub 3 for y x + 3 – 3 = 9 – 3 x = 6 Soln: {(6, 3)} 6. Check: x + y = 9 -x + y = = = -3 9 = 9 -3 = -3
add

5. Ex. Solve by the addition method: -5x + 2y = -6 10x + 7y = 34
Done
x + 2y = -6  2(-5x + 2y)=2(-6)  -10x + 4y = -12
10x + 7y = 34  10x + 7y =  10x + 7y = 34
x + 4y = -12
10x + 7y = 34
y = 22
11y = 22
y = 2
add

6. 5. 10x + 7y = 34 10x + 7(2) = 34 sub 2 for y 10x + 14 = 34 10x + 14 – 14 = 34 – 14 10x = x = 2 Soln: {(2, 2)}
Check:
-5x + 2y = x + 7y = 34
-5(2) + 2(2) = (2) + 7(2) = 34
= = 34
-6 = = 34

7. Ex. Solve by the addition method: 3x + 2y = -1 -7y = -2x – 9
Rewrite 2nd eqn. in standard form (Ax + By = C)
-7y = -2x – 9
-7y + 2x = -2x – 9 + 2x
2x – 7y = -9
2. 3x + 2y = -1  7(3x + 2y) =7(-1)  21x + 14y = -7
2x – 7y = -9  2(2x – 7y) = 2(-9)  4x – 14y = -18
x + 14y = -7
4x – 14y = -18
x + 0 = -25
25x = -25
x = -1
add

8. 5. 3x + 2y = -1 3(-1) + 2y = -1 sub -1 for x y = y + 3 = y = y = 1 Soln: {(-1, 1)}

9. Ex. Solve by the addition method: -2x = 4y + 1 2x + 4y = -1
Rewrite 1st eqn. in standard form (Ax + By = C)
-2x = 4y + 1
-2x – 4y = 4y + 1 – 4y
-2x – 4y = 1
2x + 4y = -1
x – 4y = 1
= 0
No variables remain and a TRUE stmt.
lines coincide
infinite number of solns.
dependent eqns.
Soln: {(x, y)|2x + 4y = -1}
add

10. Ex. Solve by the addition method: -3x – 6y = 4 3(x + 2y + 7) = 0
Rewrite 2nd eqn. in standard form (Ax + By = C)
3(x + 2y + 7) = 0
3x + 6y + 21 = 0
3x + 6y + 21 – 21 = 0 – 21
3x + 6y = -21
-3x – 6y = 4
x – 6y = 4
= -17
No variables remain and a FALSE stmt.
lines are parallel
no solution
inconsistent system
Answer: no soln. or ø (empty set)
add

11. Groups
Page 315 – 316: 27, 41, 59 Groups or class discussion 27 -> answer has fractions 41-> clear fractions first 59-> distribute first