Category 9 Example 1 - calculation not required
It can be observed that the values of a and b in the top equation are half of the bottom equation, but this is not true for the values of c. This implies that the ratios of a and b are same but different than that of c. Hence, the system has no solution.
Category 9 Example 2 - mental math for those good with dividing fractions
It can be observed that the values of a, b, and c in the top equation are six times of the bottom equation: 3 is six times of 1/2, 2 is six times of 1/3, and 6 is six times of 1. This implies that the ratios of a, b, and c are same. Hence, the system has infinitely many solutions.
Category 10 Example 1 - mental math
It can be observed that in the top equation the value of b is 4 times of the bottom equation. For the system to have no solution, the value of a in the top equation must also be 4 times of the bottom equation. (This will result in the same ratios of a and b.)
Since a = 1 in the bottom equation, it must be 4 in the top equation. Hence, 2n = 4 - -> n = 2.
Category 10 Example 2 - mental math
It can be observed that in the top equation the values of a and b are one-third of the bottom equation. For the system to have no solution, the value of c in the top equation cannot be one-third of the bottom equation. Hence k = c cannot be 1.
Category 11 Example 1 - mental math
It can be observed that in the top equation the value of b is 3 times of the bottom equation. Since the system has infinitely many solutions, the vales of a and c in the top equation must also be 3 times of the bottom equation (this will result in the same ratios of a, b, and c).
Hence, a = 12 and c = 3.
Category 11 Example 2 - mental math for those good with dividing fractions
It can be observed that in the top equation the value of c is 12 times of the bottom equation. Since the system has infinitely many solutions, the values of a and b in the top equation must also be 12 times of the bottom equation (this will result in the same ratios of a, b, and c).
Hence, a = 4 and b = 3.
Category 12 Example 1 - mental math
It can be observed that the value of b in the bottom equation is five times of the top equation. For the system to have one solution, the value of a = k in the bottom equation cannot be 5 times the value of a in the top equation. Hence, k cannot be 10.
Category 12 Example 3 - shortcut and may be mental math
The question asks for the value of 50x + 10y. It can be observed that if the two equations are added, the result is the value of 5x + y. The value can be multiplied by 10 to obtain the answer.
Category 14 Example 1 - mental math
Since the equation has infinitely many solutions, the coefficient of x in the left expression must be 9 and the constant in the right expression must be 6. Hence, k = 3 and c = 2.
Category 14 Example 4 - mental math
It can be observed that the constant in left and right expression is same. For the equation to have no solution, 0.5a cannot be 10. Hence, a cannot be 20.