To find which equation has the smallest slope, first convert any equations to slope intercept form (y = mx + b, where m is the slope and b is the y-intercept) if they are not already in that form. In choice E, dividing both sides of 7y = 3x − 7 by 7 yields y = \(\frac{3x}{7}\) − 1, which has a slope of ³⁄₇. When compared to the other slopes, ¹⁄₂ is the smallest (the next closest is ¹⁄₂, which is only slightly larger than ³⁄₇).

Any line parallel to the y-axis is a vertical line. Vertical lines have slopes that are undefined. Remember that the definition of slope is rise/run; vertical lines have no run and thus dividing rise by run is dividing by 0, making the quotient undefined.

To solve this problem, first distribute then combine like terms. Distribute
0.1 as follows:
0.1(t + 3,420) = t
0.1t + 342 = t.
342 = 0.9t.
t = \(\frac{342}{0.9}\) = 380

Given that |5 − 2x| > 5, then either 5 − 2x > 5 or 5 − 2x < −5. In the case that 5 − 2x > 5, −2x > 0 making x < 0 (when you divide by a negative number remember to switch the direction of the inequality). In the case that 5 − 2x < −5, then −2x < −10 making x > 5. Thus the range for x is x < 0 or x > 5. Of the answer choices, only 6 fits into the range for x.

Between May and August, there were 3 price decreases (in June, July, and August). If the price was decreased by 20%, then the resulting price was 80% of the previous month’s price. Thus, in June the price was 0.8d; in July the price was 0.8(0.8d); in August the price was 0.8(0.8(0.8d)), which is equivalent to (0.8)3d, or 0.512d.

To divide ^ax⁄_a^y , subtract the exponents. Thus ^ax⁄_a^y = a^x−y. If a^x⁄_a^y = a⁴, the a^x−y = a⁴, making x − y = 4.

To solve this problem, convert 5x + 3y = 8 to the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. To do so, first subtract 5x from both sides to get 3y = −5x + 8. Dividing the entire equation by 3 yields y = −\(\frac{5x}{3}\)+⁸⁄₃. Thus ⁸⁄₃

To solve this problem, it may be helpful to draw a picture in which one angle of a right triangle is labeled A, as shown below:

Because cos A = ⁴⁄₅, let the side adjacent to angle A be 4 and the hypotenuse be 5. Likewise, since sin A = ³⁄₅, let the side opposite angle A be 3 and the hypotenuse be 5. It follows then that tan A =side opposite divided by side adjacent, or ³⁄₄

In order to add ¹⁄_x + ³⁄₄, the fractions must have a common denominator. To achieve a common denominator, multiply ¹⁄_x by ⁴⁄₄ and ³⁄₄ by ^x⁄_x to get \(\frac{4}{4x}+\frac{3x}{4x}\), which equals \(\frac{(4 + 3x)}{4x}\)