**Category 8 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 8 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 9 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 9 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 10 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 10 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 11 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 11 Example 3** - shortcut and may be mental math

The question asks for the value of 50*x* + 10*y*. It can be observed that if the two equations are added, the result is the value of 5*x* + *y*. The value can be multiplied by 10 to obtain the answer.

**Category 13 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 13 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, 0.5*a* cannot be 10. Hence, *a* cannot be 20.

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