Answer:
-3
Explanation:
You want the multiplier of the second equation that would result in eliminating the x-terms when the first equation is multiplied by 7 and added to the multiplied second equation.
- -3x -7y = -56
- -7x +10y = 1
Multiplier
The desired multiplier will have the effect of making the coefficient of x be zero when the multiplications and addition are carried out. If k is that multiplier, the resulting x-term will be ...
7(-3x) +k(-7x) = 0
-21x -7kx = 0 . . . . . . simplify
3 +k = 0 . . . . . . . . . divide by -7x
k = -3 . . . . . . . . . . subtract 3
The multiplier of the second equation should be -3.
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Additional comments
Carrying out the suggested multiplication and addition, we have ...
7(-3x -7y) -3(-7x +10y) = 7(-56) -3(1)
-49y -30y = -395
y = -395/-79 = 5
The solution is (x, y) = (7, 5).
In general, the multipliers will be the reverse of the coefficients of the variable, with one of them negated. Here the coefficients of x are {-3, -7}. When these are reversed, you have {-7, -3}. When the first is negated, the multipliers of the two equations are {7, -3}. That is, the second equation should be multiplied by -3, as we found above.
Note that if you subtract the multiplied equations instead of adding, you can use the reversed coefficients without negating one of them. The choice of where the minus sign appears (multiplication or subtraction) will depend on your comfort level with minus signs.
The number of minus signs in this system can be reduced by multiplying the first equation by -1 to get 3x +7y = 56.
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