Practice Problems, Test #2, MAT 101

(1) Graph the linear equation:

(a) 2x - 3y = 6,

(b) y = 2x + 5

(d) x = -3.

(2) Find the slope of the line:

(a) containing (-2,8) and (1, -4)

(b) containing (3,6) and (10,6)

(d) with equation

(3) Determine whether the lines are parallel, perpendicular, or neither:

(a)

(b)

(4) (a) Find the equation of the line containing (-2, 8), with slope m = 4.

(b) Find the equation of the line containing (-2,-10) and (4, 2).

(5) A phone bill consists of a base (service fee), plus a cost for minutes used. 80 minutes of phone usage costs $20.60, and 120 minutes costs $23.40. Give a linear equation which describes the cost in terms of x = minutes used.

(6) For the linear function ,

(a) evaluate the function at f(0) and f(4)

(b) Use the slope and y-intercept to graph the function.

(7) Graph the linear inequality:

(a)

(b) y > 2

Solutions:

(1) You need to find two points on the line, graph the points, and then draw the line between them. We complete the table to find the two points:

x y

0 -2 2(0) - 3y = 6,

3 0 2x - 3(0) = 6,

The two points are (0,-2) and (3,0). Now we graph them, and draw the line between them:

(b) y = 2x + 5

x y

0 5 y = 2(0) + 5

y = 5

0 0 = 2x + 5

-5 -5

-5 = 2x

The points are (0,5) and (,0)

(d) This is the graph of a vertical line going through x = -3 on the x-axis.

(2) (a) Make sure that you know the slope formula

Here we use it like

(b)

(c) This equation is in slope-intercept form (it's solved for y), which means that we can read off the slope as the coefficient on x) .

(d) This one isn't in slope-intercept form, but we can get it in that form by solving for y.

(3) (a) We need to find the slope of both lines:

The first one is in slope-intercept form with slope m = 4. The other one needs to be solved for y:

so the second line has slope .

The slopes are not equal, but they are negative inverses, and so they are perpendicular.

(b)

Find the slopes of the lines by solving both for y:

The slopes are both , thus the lines are parallel.

(4)(a) I'm going to use point-slope form , with

(b) I'm going to use point-slope form again, but this time I need to find the slope first:

, and then we need to pick one of the points to be = (4,2), and then:

(5) Write the data as ordered pairs, with x = the number of minutes, and y = the cost:

(80, 20.6) and (120, 23.4) , the find the slope of the line:

, then using point-slope form:

I picked =(120, 23.4), so:

(6) (a)

(b) The y-intercept is (0,-2), and we use the slope to find the second point – go right 4, and 3 up, and graph the second point, then draw the line between them.

(7) First, graph the border line 2x + 3y = 6:

x y

0 2

3 0

So, graph (0,2), and (3,0) , and the line between them.

Then test a point to see whether it is in the solution region - I'm going to use (0,0), like I always do (if I can - i.e. if the origin does not lie on the border line).

2(0) + 3(0) < 6?

0 < 6, so yes, the origin is in the solution region - so shade the half of the real plane on the side of the border line where (0,0) is:

(b) The border will be the horizontal line y = 2, then test (0,0) : 0 > 2? no, so graph the side of the line above it (where the origin isn't)

(c) Graph the vertical line x = 3, then test the origin: 0 < 3 yes, so shade the sign of the line where the origin is (on the left):