Transcript 5.3 Solving Systems of Linear Equations by the Addition

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. Done
2. x + y = 3
x–y=5
3. 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
add
Soln: {(4, -1)}
6. Check:
x+y=3
4 + (-1) = 3
3=3
x–y=5
4 – (-1) = 3
4+1=5
5=5
Ex. Solve by the addition method: x + y = 9
-x + y = -3
1. Done
2. x + y = 9
-x + y = -3
3. x + y = 9
-x + y = -3
4. 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
add
Soln: {(6, 3)}
6. Check:
x+y=9
6+3=9
9=9
-x + y = -3
-6 + 3 = -3
-3 = -3
Ex. Solve by the addition method: -5x + 2y = -6
10x + 7y = 34
1. Done
2. -5x + 2y = -6  2(-5x + 2y)=2(-6)  -10x + 4y = -12
10x + 7y = 34  10x + 7y = 34  10x + 7y = 34
3. -10x + 4y = -12
add
10x + 7y = 34
4.
0 + 11y = 22
11y = 22
11y = 22
11 11
y=2
5.
10x + 7y = 34
6. Check:
10x + 7(2) = 34 sub 2 for y
-5x + 2y = -6
10x + 7y = 34
10x + 14 = 34
-5(2) + 2(2) = -6 10(2) + 7(2) = 34
10x + 14 – 14 = 34 – 14
-10 + 4 = -6
20 + 14 = 34
10x = 20
-6 = -6
34 = 34
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 + 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
3. 21x + 14y = -7 add
4x – 14y = -18
4. 25x + 0 = -25
25x = -25
25x = -25
25
25
x = -1
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
No variables remain and a TRUE stmt.
2. -2x – 4y = 1
lines coincide
2x + 4y = -1
infinite number of solns.
3. -2x – 4y = 1
dependent eqns.
add
2x + 4y = -1
Soln: {(x, y)|2x + 4y = -1}
4. 0 + 0 = 0
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 = -21
No variables remain and a FALSE stmt.
2. -3x – 6y = 4
lines are parallel
3x + 6y = -21
no solution
3. -3x – 6y = 4
inconsistent system
add
3x + 6y = -21
Answer: no soln. or ø (empty set)
4.
0 + 0 = -17
Groups
Page 315 – 316: 27, 41, 59
Groups or class discussion
27 -> answer has fractions
41-> clear fractions first
59-> distribute first