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

$Equation 1: 2 x, plus 7 y, equals 9. Equation 2: 8 x, plus 28 y, equals a$

In the given system of equations, a is a constant. If the system has infinitely many solutions, what is the value of a ?

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Explanation

Choice C 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 of the equation and a constant on the right-hand side of the equation. The coefficients of x and y in the second equation are equal to the coefficients of x and y, respectively, in the first equation multiplied by 4: $8 equals, 2 times 4$ and $28 equals, 7 times 4$ . Therefore, the constant in the second equation must be equal to 4 times the constant in the first equation: $a, equals, 9 times 4$ , or $a, equals 36$ .

Choices A, B, and D are incorrect. When $a, equals 4$ , $a, equals 9$ , or $a, equals 54$ , the given system of equations has no solution.