solve by elimination method to find an ordered pair.
7x-y=27
x+9y=77
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7 x - y = 27 Divide both sides by 7
7 x / 7 - y / 7 = 27 / 7
x - y / 7 = 27 / 7
x + 9 y = 77
x + 9 y * 7 / 7 = 77 * 7 / 7
x + 63 y / 7 = 539 / 7
Subtract x - y / 7 = 27 / 7 from equation x + 63 y / 7 = 539 / 7
x + 63 y / 7 = 539 / 7
- x - y / 7 = 27 / 7
________________________
x - x + 63 y / 7 - ( - y / 7 ) = 539 / 7 - 27 / 7
0 + 63 y / 7 + y / 7 = 512 / 7
64 y / 7 = 512 / 7 Multiply both sides by 7
64 y = 512 Divide both sides by 64
64 y / 64 = 512 / 64
y = 8
Substitute x in either of the original equations to get the value of y.
7 x - y = 27
7 x - 8 = 27 Add 8 to both sides
7 x - 8 + 8 = 27 + 8
7 x = 35 Divide both sides by 7
7 x / 7 = 35 / 7
x = 5
Solution :
x = 5 , y = 8
Or an ordered pair ( 5 , 8 )
OR
x + 9 y = 77 Multiply both sides by 7
7 * x + 9 y * 7 = 77 * 7
7 x + 63 y = 539
-
7 x - y = 27
_______________
7 x - 7 x + 63 y - ( - y ) = 539 - 27
0 + 63 y + y = 512
64 y = 512
y = 512 / 64 = 8
x + 9 y = 77
x + 9 * 8 = 77
x + 72 = 77 Subtract 72 to both sides
x + 72 - 72 = 77 - 72
x = 5
( 5 , 8 )
To solve the system of equations using the elimination method, we want to eliminate one variable by scaling one or both equations, then add or subtract the equations to eliminate the variable.
Given equations:
1) 7x - y = 27
2) x + 9y = 77
Step 1: Multiply equation 1 by 9 and equation 2 by -1 to make the coefficients of y the same.
(9) * (7x - y) = (9) * (27) => 63x - 9y = 243
(-1) * (x + 9y) = (-1) * (77) => -x - 9y = -77
The new equations become:
3) 63x - 9y = 243
4) -x - 9y = -77
Step 2: Add equation 3 and equation 4 to eliminate the variable y.
(3) + (4): (63x - 9y) + (-x - 9y) = 243 - 77
=> 62x - 18y = 166
So far, we have the equation:
5) 62x - 18y = 166
Step 3: Solve equation 5 for x.
62x - 18y = 166
62x = 166 + 18y
x = (166 + 18y) / 62
Step 4: Substitute the value of x in equation 1 to find y.
7x - y = 27
7((166 + 18y) / 62) - y = 27
Multiply both sides by 62 to eliminate the fraction:
7(166 + 18y) - 62y = 27 * 62
Distribute:
1162 + 126y - y = 1674
125y = 1674 - 1162
125y = 512
y = 512 / 125
y = 4.096 (approximately)
Step 5: Substitute the value of y back into equation 1 to find x.
7x - y = 27
7x - 4.096 = 27
7x = 27 + 4.096
7x = 31.096
x = 31.096 / 7
x = 4.442 (approximately)
Therefore, the ordered pair solution is (x, y) ≈ (4.442, 4.096).