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8th Grade
Chapter 8 – Solving Systems of Equations
8.3 By “The Addition Method”: Page 367
8.3 Solve by the “Addition Method”:
This technique for solving a system of equations is
useful when they are written in the Standard Form of
Ax + By = C.
x + y = 5 and
x–y=1
The Addition Property of Equality. We can add the
same number to either side and the Equations are still
“balanced” or equal.
X + y = 5
+x – y = 1
2x + 0y = 6
2x = 6
x=3
Substitute back into either of the equations:
Check in the 2nd equation:
x – y = 1:
Try This: page 367
a) x + y = 5 and
2x - y = 4
x + y = 5:
3–2=1
3+y=5
1 = 1 Valid
∴y=2
b) 3x - 3y = 6 and 3x + 3y = 0
The Multiplication Property of Equality is we can multiply each side of an equation by the same number
and still have a balanced or equal equation.
2x + 3y = 8
and x + 3y = 7
In the above example, no variables will be eliminated by just adding the terms, BUT, if we multiple one
of the equations by (-1) we create a Linear Combination and additive inverse.
2x + 3y = 8
2x + 3y = 8
and (-1)(x + 3y) = (-1)(7)
+
-x – 3y = -7
x + 0y = 1
Substitute back into the equation:
x + 3y = 7:
1 + 3y = 7:
3y = 6 :
y=2
Check in the other equation:
2x + 3y = 8:
2(1) + 3(2) = 8:
2+6=8
Try This: page 368:
c) 5x + 3y =17
and
d) 8x + 11y = 37 and
5x – 2y = -3
-2x + 11y = 7
Sometimes you can find a Multiplicative Inverse to eliminate a term:
3x + 6y = -6 and
5x – 2y = 14
The “Y” term of -2 can be multiplied by 3. So:
3(5x – 2y) = (3)(14)
15x – 6y = 42
3x + 6y = -6
+ 15x – 6y = 42
18x + 0y = 36
18x = 36
x=2
8th Grade
Chapter 8 – Solving Systems of Equations
8.3 By “The Addition Method”: Page 367
Try This page: 369
e) 4a + 7b = 11
f) 7x – 5y = 76
g) 5b + 10c – 15
and
and
and
4a + 6b = 10
4x + y = 55
3b – 2c = -7
The Multiplication Property can be used multiple times to FORCE the elimination of a term by creating
the additive inverse:
3x + 5y = 30
and
5x + 8y = 49
5(3x + 5y) = (5)(30)
15x + 25y = 150
and
and
(-3)(5x + 8y) = (-3)(49)
-15x – 24y = -147
15x + 25y = 150
+ -15x – 24y = -147
0x + y = 3
Substitute back into the equation:
5x + 8y = 49
5(x) + 8(3) = 49
5(x) + 24 = 49
5x = 25
x=5
Try This page 369
h) 5x + 3y = 2
i) 6x + 2y = 4
and
and
3x + 5y = -2
10x + 7y = -8
The Multiplication Property can be used eliminated fractional coefficients to create additive inverse terms:
𝟏
𝟏
x + y = 56
and
𝒙 + 𝟒 𝒚 = 𝟏𝟔
𝟑
𝟏
𝟏
12(𝟑 𝒙 + 𝟒 𝒚) = (𝟏𝟐)𝟏𝟔
-4(x _+ y) = (-4)(56)
so……
4x + 3y = 192
- 4x – 4y = - 224
0x – y = -32
y = 32
8th Grade
Chapter 8 – Solving Systems of Equations
8.3 By “The Addition Method”: Page 367
Exercises 8-3 page 371
1) x + y = 10
and
x–y=8
2) x – y = 7
and
x+y=3
3) x + y = 8
and
– x + 2y = 7
4) x + y = 6
and
–x + 3y = -2
5)3x – y = 9
and
2x + y = 6
6) 4x – y = 1
and
3x + y = 13
7) 4a + 3b = 7 and
-4a + b = 5
8) 7c + 5d = 18 and
c – 5d = -2
9) 8x – 5y = -9 and
3x + 5y = -2
10) 3a – 3b = -15 and -3a – 3b = -3
11) 4x – 5y = 7 and
-4x + 5y = 7
12) 2x + 3y = 4 and
-2x – 3y = -4
13) –x – y =8 and
2x – y = -1
14) x + y = -7 and
3x + y = -9
15) x + 3y =19 and
x – y = -1
16) 3x – y =8 and
x + 2y =5
17) x + y = 5
5x – 3y = 17
18) x – y = 7
4x – 5y -25
19) 2w + 3z =17
and
3w + 4z = 24
20) 7p + 5q =2 and
8p +-9q =17
21) 2a + 3b = -1
and
3a + 5b = -2
22) 3x – 4y =16 and
5x + 6y = 14
23) x – 3y = 0
and
5x – y = -14
24) 5a – 2b =0 and
2a – 3b = -11
25) 3x – 2y = 10
and
5x + 3y = 4
26) 2p + 5q = 9 and
3p – 2q = 4
27) 3x – 8y = 11
and
x + 6y -8 = 0
28) m – n = 32 and
3m = 8n – 6 = 0
29) a + b =12
and
½ a + ¼b = 4
30) 2p – q = 8 and
1/3p + 1/4q = 3
and
and
Translate:
31) The sum of two numbers is 115 and the difference is 21.
32) The sum of two numbers is 26.4. One is five times the other.
33) The sum of the length and width of a rectangle is 19 inches. The length is one less than
twice the width.
34) The perimeter of a rectangle is 48m. The width of the rectangle is 2 more than half the length
35) Two angles are complementary. Their difference is 34o.
36) Two angles are complementary. One angle is 420 more than ½ the other.
38) 3(x – y) = 9
and
x+y=7
39) 5(a – b) = 10
and a + b = 2
40) 2(x – y) = 3 + x and
x = 3y + 4
41) 2(5a – 5b) =10 and -5(6a + 2b) =10
42) 1.5x + .85y = 1637.5
and .01(x + y) = 15.25
44) y = ax + b
and y = x + c
45) ax + by + c = 0 and ax + cy + b = 0
46) 2(7-a) – 2(1 + 2b) + 5 = 0 and 2a + 2b -18 = 0
47) 2/x– 3/y = - ½
and 1/x + 2/y = 11/12