Solutions to systems of equations and perpendicular line problems

Mathematics

Determine whether the lines will be perpendicular when graphed.

For two lines to be perpendicular, the slope of one line must be the negative reciprocal of the slope of the other line. The slope can be determined by transforming the equation to its standard form of y = mx + b, where m = slope. Thus,

2y = 6 2y = 3x - 6 3/2x - 3, where m = 3/2

For the other line to be perpendicular, the slope of the equation for 2x + 3y = 6 MUST be m = -2/3. To check this, the equation can be transformed to its standard form:

3y = 6 3y = 6-2x y = 2-2/3x y = -2/3x + 2, where m = -2/3

Therefore, the lines of both equations will be perpendicular to each other when graphed.

Alice's Restaurant has a total of 205 seats. The number of seats in the non-smoking section is 73 more than twice the number of seats in the smoking section. How many seats are in each section?

Let x = non-smoking seats, and y = smoking seats, with the total of x + y = 205.

From the conditions listed above, x = 2y + 73. Substituting for x in the original equation, we get 205 = 2y + 73 + y 205 = 3y + 73-132 = 3y y = 44, x = 161.

Therefore, there are a total of 161 non-smoking seats and 44 smoking seats in Alice's Restaurant.

3. Solve the system of equations by the substitution method

Equation 1: x + 3y = 32, Equation 2: -3x + 2y = 3

Equation 1: x + 3y = 32 x = 32-3y

Substituting x into Equation 2, we get

-3x + 2y = 3 -3(32-3y) + 2y = 3 -96 + 9y + 2y = 3 11y = 99 y = 9

Plugging back into Equation 1, we get x + 3(9) = 32 x + 27 = 32 x = 5

Therefore, x = 5, y = 9

4. Solve the system of equations

Equation 1: x + y = 5, Equation 2: x - y = -9

Equation 1: x + y = 5 x = 5 - y

Substituting x into Equation 2, we get x - y = 9-5 - y - y = -9 -2y = -14 y = 7

Plugging back into Equation 2, we get x + y = 5 x + 7 = 5 x = -2

Therefore, x = -2, y = 7