5.3 Solving Systems of Linear Equations by the Addition Method
Solving Using Addition Method
1. Write both eqns. in standard form (Ax + By = C).2. Get opposite coefficients for one of the variables.
You may need to mult. one or both eqns. by a nonzero number to do this.
3. Add the eqns., vertically.4. Solve the remaining eqn. 5. Substitute the value for the variable from step 4
into one of the original eqns. and solve for the other variable.
6. Check soln. in BOTH eqns., if necessary.
Ex. Solve by the addition method: x + y = 3 x – y = 5
1. Done2. x + y = 3
x – y = 53. x + y = 3
x – y = 5 4. 2x + 0 = 8 2x = 8
2x = 8 2 2
x = 4
5. x + y = 3 4 + y = 3 sub 4 for x y + 4 – 4 = 3 – 4
y = -1
Soln: {(4, -1)}
6. Check: x + y = 3 x
– y = 54 + (-1) = 3 4 – (-1) = 3
3 = 3 4 + 1 = 5
5 = 5
add
Ex. Solve by the addition method: x + y = 9 -x + y = -3
1. Done2. x + y = 9 -x + y = -33. x + y = 9
-x + y = -34. 0 + 2y = 6 2y = 6
2y = 6 2 2
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 6 + 3 = 9 -6 + 3 = -3
9 = 9 -3 = -3
add
Ex. Solve by the addition method: -5x + 2y = -6 10x + 7y = 34
1. Done2. -5x + 2y = -6 2(-5x + 2y)=2(-6) -10x + 4y = -12 10x + 7y = 34 10x + 7y = 34 10x + 7y = 343. -10x + 4y = -12
10x + 7y = 344. 0 + 11y = 22 11y = 22
11y = 22 11 11
y = 2
add
6. Check: -5x + 2y = -6 10x + 7y = 34-5(2) + 2(2) = -6 10(2) + 7(2) = 34 -10 + 4 = -6 20 + 14 = 34 -6 = -6 34 = 34
5. 10x + 7y = 34 10x + 7(2) = 34 sub 2 for y
10x + 14 = 34 10x + 14 – 14 = 34 – 14
10x = 20 10x = 20
10 10 x = 2
Soln: {(2, 2)}
Ex. Solve by the addition method: 3x + 2y = -1 -7y = -2x – 9
1. Rewrite 2nd eqn. in standard form (Ax + By = C)-7y = -2x – 9 -7y + 2x = -2x – 9 + 2x2x – 7y = -9
2. 3x + 2y = -1 7(3x + 2y) =7(-1) 21x + 14y = -7 2x – 7y = -9 2(2x – 7y) = 2(-9) 4x – 14y = -183. 21x + 14y = -7 4x – 14y = -184. 25x + 0 = -25 25x = -25
25x = -25 25 25
x = -1
add
5. 3x + 2y = -1 3(-1) + 2y = -1 sub -1 for x
-3 + 2y = -1 -3 + 2y + 3 = -1 + 3
2y = 2 2y = 2 2 2
y = 1
Soln: {(-1, 1)}
Ex. Solve by the addition method: -2x = 4y + 1 2x + 4y = -1
1. Rewrite 1st eqn. in standard form (Ax + By = C)-2x = 4y + 1-2x – 4y = 4y + 1 – 4y -2x – 4y = 1
2. -2x – 4y = 1 2x + 4y = -1
3. -2x – 4y = 1 2x + 4y = -14. 0 + 0 = 0
add
No variables remain and a TRUE stmt. lines coincideinfinite number of solns.dependent eqns.
Soln: {(x, y)|2x + 4y = -1}
Ex. Solve by the addition method: -3x – 6y = 4 3(x + 2y + 7) = 0
1. 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 = -212. -3x – 6y = 4
3x + 6y = -21 3. -3x – 6y = 4 3x + 6y = -214. 0 + 0 = -17
add
No variables remain and a FALSE stmt. lines are parallelno solutioninconsistent system
Answer: no soln. or ø (empty set)
Groups
Page 315 – 316: 27, 41, 59
Groups or class discussion27 -> answer has fractions41-> clear fractions first59-> distribute first
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