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Algebra / Systems of two linear equations in two variables Difficulty: Hard

In the system of equations below, a and c are constants.

$Equation 1: one half x, plus one third y, equals one sixth. Equation 2: a, x, plus y equals c$

If the system of equations has an infinite number of solutions $x comma y$ , what is the value of a ?

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Explanation

Choice D is correct. A system of two linear equations has infinitely many solutions if one equation is equivalent to the other. This means that when the two equations are written in the same form, each coefficient or constant in one equation is equal to the corresponding coefficient or constant in the other equation multiplied by the same number. The equations in the given system of equations are written in the same form, with x and y on the left-hand side and a constant on the right-hand side of the equation. The coefficient of y in the second equation is equal to the coefficient of y in the first equation multiplied by 3. Therefore, a, the coefficient of x in the second equation, must be equal to 3 times the coefficient of x in the first equation: $a, equals, one half times 3$ , or $a, equals three halves$ .

Choices A, B, and C are incorrect. When $a, equals negative one half$ , $a, equals 0$ , or $a, equals one half$ , the given system of equations has one solution.