MATH 1010-2: Quiz 4A
Solution
1. (a) (3 points) Find the slope of the line that passes through the two points (1, 2) and (3, −2).
(b) (2 points) Write an equation of the line in part (a).
Solution: We will set the two points (x1 , y1 ) = (1, 2) and (x2 , y2 ) = (3, −2). By the slope formula,
slope =
y2 − y 1
(−2) − 2
−4
=
=
= −2
x2 − x1
3−1
2
So the slope of this line is −2.
To write an equation, we are going to use the point-slope form, since we have the slope which is
−2 and one point on this line which we choose to be (1, 2) (you may also choose (3, −2) which will
give the same equation). The equation is
y − 2 = (−2)(x − 1)
Simplifying it, we have
y − 2 = −2x + 2
y = −2x + 4
To conclude, the equation of this line is y = −2x + 4.
2. (5 points) Graph 2x + 5y > 10 in a rectangular coordinate system.
Solution: The first step is to find the graph of the corresponding equation, 2x + 5y = 10. Since
(5, 0) and (0, 2) satisfy this equation, the graph should be the line through these two points as in
the picture below. The symbol of inequality is > here, so the line should be dashed.
To determine which side of the line is the graph of 2x + 5y > 10, we will pick (0, 0) since it doesn’t
lie on this line and check if it satisfies the inequality.
LHS > RHS
2 × 0 + 5 × 0 > 10
0 > 10
This inequality is false. We have (0, 0) is not a solution of the inequality, so the graph should be
the other side of the line.
y
3
2
1
b
b
b
b
0
b
1
b
b
b
b
2
3
4
5
x
MATH 1010-2: Quiz 4B
Solution
1. (a) (3 points) Find the slope of the line that passes through the two points (−1, −2) and (3, 2).
(b) (2 points) Write an equation of the line in part (a).
Solution: We will set the two points (x1 , y1 ) = (−1, −2) and (x2 , y2 ) = (3, 2). By the slope formula,
y2 − y 1
2 − (−2)
4
=
= =1
x2 − x1
3 − (−1)
4
slope =
So the slope of this line is 1.
To write an equation, we are going to use the point-slope form, since we have the slope which is 1
and one point on this line which we choose to be (−1, −2) (you may also choose (3, 2) which will
give the same equation). The equation is
y − (−2) = 1 · (x − (−1))
Simplifying it, we have
y+2=x+1
y =x−1
To conclude, the equation of this line is y = x − 1.
2. (5 points) Graph 3x + 4y < 12 in a rectangular coordinate system.
Solution: The first step is to find the graph of the corresponding equation, 3x + 4y = 12. Since
(4, 0) and (0, 3) satisfy this equation, the graph should be the line through these two points as in
the picture below. The symbol of inequality is < here, so the line should be dashed.
To determine which side of the line is the graph of 3x + 4y < 12, we will pick (0, 0) since it doesn’t
lie on this line and check if it satisfies the inequality.
LHS < RHS
3 × 0 + 4 × 0 < 12
0 < 12
This inequality is true. We have (0, 0) is a solution of the inequality, so the graph should be this
side of the line.
y
3
2
1
b
b
b
b
0
b
1
b
b
b
b
2
3
4
5
x