International projectionist (Jan-Dec 1939)
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number without changing the equation's value, and we also know that we can divide both sides of any equation by any chosen number without changing the equation's value. In such operations we of course change the magnitude of individual coefficients.
Thus when we are presented, with simultaneous equations which do not have equal coefficients for some unknown which we wish to eliminate, , we simply multiply or divide one or both equations by chosen numbers so that equal coefficients for the particular unknown are formed in both equations. Consider the appended simultaneous equations Nos. 1 and 2:
1. 3x+7y = -23
2. 2x + v = 3
We note that we can cause the coefficient of x to be 6 in both equations if we multiply equation No. 1 throughout by 2. and No. 2 throughout by 3. We can then eliminate by subtraction:
No. 1X2. 6x+14y = -46 No. 2X3. 6x + 3v = 9
llv
-55
y = —5 Place y = — 5 in equation No. 2: 2x+y = 3 2x+ (-5) = 3 2x-5 = 3 2x = 3+5 2x = 8 x = 1
Thus x = 4 and y = — 5 is the solution. For a ready check, place these values in equation No. 1. If these values satisfy equation our solution is correct.
1. 3sc+7y = -23
(3) (4) + (7) (-5) = -23
12+ (-35) = -23
12-35 = -23
-23 = -23 check
Appended hereto is another example, this one showing how an equation may be divided throughout by a number so that we can proceed with a solution by addition:
1. 21x+7y = 336
2. -3x+ y = 6
Divide No. 1 by 7, and add equation No. 2: 3x+ y = 48 -3x+ v = 6 2y = 54 y = 27 Place y — 27 in equation No. 2: -3x+ y = 6 -3x+27 = 6
-3x = 6-27 -3x = -21
To check, place x = 7 and y = 27 in equasion No. 1:
21x+7y = 336
(21) (7) + (7) (27) = 336
147 + 189 = 336
336 = 336 check
2. Elimination by substitution of values.
In this method we take one of the equations and transpose terms until we obtain an expression for one of the unknowns. We substitute this expression in the other equation. By this substitution one of the unknowns is eliminated in the other equation. This will be clarified if we work out an actual example. Consider the appended simultaneous equations Nos. 1 and 2:
1. x+2y = 10
2. 2x+ y = 8 Solve for x from equation No. 1:
x = 10 -2y
Substitute this value of x in equation No. 2 and simplifv:
2x+ y = 8
2(10-2y)+y = 8
20-4y+ y = 8
-3y = 8-20
-3y = -12 y = 4
Place y = 4 in equation No. 2:
2x+y = 8 2x+4 = 8
2x = 8-4
2x = 4
x = 2
Tc check, place y equation No. 1:
4 and x = 2 ia
x+2y = 10
2 +(2) (4) = 10
2+8 = 10
10 = 10
Appended is the solution of another example using the substitution method, but this time we have omitted most of the interpretative phrasing on the sidelines. It will be good practice to follow through the steps for yourself without benefit of cues.
1.
3x-4y = -3
2.
4x+ y = 15
3x = -3 + 4v
-3+4v X= 3
2.
4x+ v = 15
<■
-3+4y\
3 ;+-v-15
-12+16-v+ „ „
(Hint: multiply throughout by 3 to clear the fraction) :
-12 + 16y+3y = 45
19 y = 45 + 12
19y = 57
y = 3
2. 4x+y = 15
4x+3 = 15
4x = 15-3
4x = 12
x = 3
1. 3x-4y = -3
(3) (3) -(4) (3) = -3
9-12 = -3
— 3 = —3 check
Hence: x = 3, y = 3
Let's apply this substitution method to a practical problem. A gun was fired, and the speed of the sound with the wind was 1070 feet per second, and 1030 feet per second against the wind. Find the velocity of sound in still air and the velocity of the wind.
We have two unknowns, which we shall call x and y.
Let x = velocity of sound in still air.
Let y = velocity of the wind.
The measured velocity, 1070 feet per second, represents the velocity of sound in still air plus the velocity of the wind: x + y = 1070.
The measured velocity, 1030 feet per second, represents the velocity of sound in still air minus the velocity of the wind: x — y — 1030. Hence our simultaneous equations are:
1. x+y = 1070
2. x-y = 1030
Solving these by the substitution method:
x = 1030 +y
1. x+ y = 1070 (1030 +y)+ y = 1070
1030 +2y = 1070
2y = 1070-1030 2y = 1040 y = 20 ft. per sec.
2. x y = 1030 x-20 = 1030
x = 1030+20 x = 1050 ft. per sec. 1. x+ y = 1070
1050+20 = 1070
1070 = 1070 check
3. Elimination by equating equal expressions.
In this method we take one of the equations and transpose terms until we obtain an expression from one of the unknowns. Then we take the other equation and transpose terms until we
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INTERNATIONAL PROJECTIONIST