Two-Variable (Simultaneous) Equations: Elimination
Consider the system of equations
$$\displaystyle ax + 3y = 43$$
$$\displaystyle -2x + by = 25$$
where _a_ and _b_ are constants.
Which of the below values for _a_ and _b_ result in a system of equations with a solution in which _x_ = 4?
Incorrect.
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By replacing each of _a_ and _b_ with 0, you obtain the system
$$\displaystyle 3y = 43$$
$$\displaystyle -2x = 25$$
$$\displaystyle x = -\frac{25}{2}$$.
Since the value of _x_ is not 4, this answer choice can be eliminated.
Incorrect.
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By replacing _a_ with 2 and _b_ with -6, you obtain the system
$$\displaystyle 2x + 3y = 43$$
$$\displaystyle -2x - 6y = 25$$.
Multiplying the first equation by 2 yields
$$\displaystyle 4x + 6y = 86$$
$$\displaystyle -2x - 6y = 25$$.
Adding those equations results in
$$\displaystyle 2x= 111$$
$$\displaystyle x = \frac{111}{2}$$.
Since the value of _x_ is not 4, this answer choice can be eliminated.
Correct.
You can solve this problem using __Reverse Plugging In__. By replacing _a_ with 8 and _b_ with 9, you obtain the system
$$\displaystyle 8x + 3y = 43$$
$$\displaystyle -2x + 9y = 25$$.
Multiplying the first equation by 3 yields
$$\displaystyle 24x + 9y = 129$$
$$\displaystyle -2x + 9y = 25$$.
Subtracting the second equation from the first results in
$$\displaystyle 26x = 104$$
$$\displaystyle x = \frac{104}{26} = 4$$.
Therefore, _a_ = 8 and _b_ = 9 provides a system with a solution in which _x_ = 4.
Incorrect.
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By replacing _a_ with 10 and _b_ with 6, you obtain the system
$$\displaystyle 10x + 3y = 43$$
$$\displaystyle -2x + 6y = 25$$.
Multiplying the first equation by 2 yields
$$\displaystyle 20x + 6y = 86$$
$$\displaystyle -2x + 6y = 25$$.
Subtracting the second equation from the first results in
$$\displaystyle 22x = 61$$
$$\displaystyle x = \frac{61}{22}$$.
Since the value of _x_ is not 4, this answer choice can be eliminated.
Incorrect.
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By replacing _a_ with 4 and _b_ with -12, you obtain the system
$$\displaystyle 4x + 3y = 43$$
$$\displaystyle -2x - 12y = 25$$.
Multiplying the first equation by 4 yields
$$\displaystyle 16x + 12y = 172$$
$$\displaystyle -2x - 12y = 25$$.
Adding these equations results in
$$\displaystyle 14x = 197$$
$$\displaystyle x = \frac{197}{14}$$.
Since the value of _x_ is not 4, this answer choice can be eliminated.
_a_ = 0, _b_ = 0
_a_ = 4, _b_ = -12
_a_ = 2, _b_ = -6
_a_ = 8, _b_ = 9
_a_ = 10, _b_ = 6
