2x + 3y = 7 ax + 9y = b In the system of equations, a and b... | SAT Systems of two linear equations in two variables Question, Answer Explained

Question bank / Systems of two linear equations in two variables

Math · AlgebraHard

$2x + 3y = 7$
$ax + 9y = b$
In the system of equations, $a$ and $b$ are constants. For which of the following values of $a$ and $b$ does the system have no solution?

$a = 2$ and $b = 7$

$a = 3$ and $b = 21$

$a = 6$ and $b = 7$

$a = 6$ and $b = 21$

Show answer and explanation

Answer: C

The system has no solution when the lines are parallel but not the same. Multiplying the first equation by $3$ gives $6x + 9y = 21$ . The second equation has the same $y$ -coefficient, $9$ , so its line is parallel to the first when $a = 6$ . If $b = 21$ as well, the two equations describe the same line. So the system has no solution when $a = 6$ and $b \ne 21$ , which is choice C.

Choice D makes the second equation exactly $3$ times the first, so the system has infinitely many solutions rather than none.

Why the other choices are wrong

A. With a = 2, the slopes are -2/3 and -2/9, so the lines intersect at exactly one point.
B. With a = 3, the slopes are -2/3 and -1/3, so the lines intersect at exactly one point.
D. With a = 6 and b = 21, the second equation is 3 times the first, so the system has infinitely many solutions.

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