Intermediate Algebra Problems With Answers - Sample 7 - Slopes of Lines
A set of intermediate algebra problems, related to slopes of lines, with answers, are presented. The solutions are at the bottom of the page.
Find the slopes of the lines given by the following equations
a) 2x + 3y = 2
b) y = 2
c) x = 4
Find the slopes of the lines through the points A and B given by
a) A(2 , 1) , B(3 , 4)
b) A(3 , -4) , B(5 , -4)
c) A(2 , -6) , B(2 , -7)
Find the slopes of the lines
a) parallel to the line whose equation is given by 5x - 3y = -3
b) perpendicular to the line whose equation is given by -4x - 8y = -3
c) parallel to the line whose equation is given x = 2
d) perpendicular to the line whose equation is given by x = -9
e) parallel to the line whose equation is given y = 3
f) perpendicular to the line whose equation is given by x = 0
Find the slope of each of the lines graphed below.
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Find the value of k so that the slope of the line through the points (4 , 2) and (k , 6) is equal to -2.
Find the value of k so that the lines with equations 2y + k x = 2 and 3x - 2y = 6 have equal slopes.
Find the value of k so that the lines with equations -3 y + 2 x = 4 and k x + 2y = 3 are perpendicular.
The oil consumption of a certain country was 330 thousands barrels per day in 2004 and 450 thousands barrels per day in 2006. Assume that the oil consumption in this country increases linearly and estimate the oil consumption in 2015.
A family spent $3600 on food last year and $3000 the year before the last. Assume that the spending on food of this family increases linearly and estimate their spending on food this year.
To convert the measure of temperature given in degree Fahrenheit T_{f} into degree Celcius T_{c} you may use the formula given by
T_{c} = (5 / 9)(T_{f} - 32)
If the temperature of an item increases by 9 degree Fahrenheit, by how many degrees Celcius has the temperature of this item increased?
To convert the measure of temperature given in degree Celcius T_{c} into degree Fahrenheit, you may use the formula given by
T_{f} = (9 / 5)T_{c} + 32
If the temperature of an item increases by 10 degree Celcius, by how many degrees Fahrenheit has the temperature of this item increased?
Answers to the Above Questions
a) Solve equation for y: y = (-2/3)x + 2/3 , slope = -2/3
b) The slope of a horizontal line such as y = 2 is equal to 0.
c) The slope of a vertical line such as x = 4 is undefined.
a) slope = (4 - 1) / (3 - 2) = 3
b) slope = (-4 - (-4)) / (5 - 3) = 0 , the line through A(3 , -4) and B(5 , -4) is a horizontal line.
c) slope = (-7 - (-6)) / (2 - 2) = undefined , the line through A(2 , -6) and B(2 , -7) is a vertical line.
a) The line 5x - 3y = -3 has slope equal to 5/3 and the line parallel to it will have equal to 5/3. (parallel lines have equal slopes)
b) The line -4x - 8y = -3 has slope equal to -1/2 and the line perpendicular to it will have equal to 2. (the product of the slopes of two perpendicular lines is equal to -1)
c) x = 2 is a vertical line. The line parallel to a vertical line is also vertical and its slope is undefined.
d) x = -9 is a vertical line and the line perpendicular to it is horizontal and its slope is equal to 0.
e) y = 3 is a horizontal line and a parallel to it is also horizontal and its slope is equal to 0.
f) x = 0 is a vertical line and the perpendicular to it is horizontal and therefore its slope is equal to 0.
a) L1 is a vertical line and its slope is undefined.
b) L2 is a horizontal line and its slope is equal to 0.
c) the slope of L3 may be found using the x and y intercepts: slope = (4 - 0) / (0 - (-4)) = 1
Slope: - 2 = (6 - 2)/(k - 4)
Solve for k
k = 2
Write equation 2y + k x = 2 in slope intercept form and find its slope m.
2 y = - k x + 2
y = - (k/2) x + 1/2
slope m = - k / 2
Write equation 3x - 2y = 6 in slope intercept form and find its slope s.
- 2 y = - 3 x + 6
y = (3/2) x - 3
slope s = 3/2
equal slopes: - k / 2 = 3 / 2
Solve above equation for k:
k = - 3
Write -3y + 2x = 4 in slope intercept form and find its slope m.
- 3 y = - 2x + 4
y = (2/3) x - 4/3
slope m = 2 / 3
Write k x + 2y = 3 in slope intercept form and find its slope s.
2 y = - k x + 3
y = (-k/2) x + 3/2
slope s = -k/2
Perpendicular lines: m * s = - 1
Substitute m and s
(2 / 3)*(-k/2) = -1
Solve for k
k = 3
Let t be the number of years after 2004; hence t = 0 in 2004. The oil consumption C increases linearly with time t which may be written as:
C = m t + C0
in 2004 corresponding to t = 0 , C = 330
Hence the equation
330 = m * 0 + C0 which give C0 = 330
C = m t + 330
The year 2006 corresponding t = 2006 - 2004 = 2, C = 450; hence
450 = m (2) + 330
m = (450 - 330) / 2 = 60
The year 2015 corresponds to t = 2015 - 2004 = 11; hence the consumption in 2015 is given by
C = 60 * 11 + 330 = 990 thousands barrels per day
The increase per year is
3600 - 3000 = $600
Spending this year
3600 + 600 = $4200
T_{c} = (5 / 9)(T_{f} - 32) = (5/9) T_{f} - 32*5/9
T_{c} varies linearly as a function of T_{f}
Hence an increase of ΔT_{f} = 9 degree Fahrenheit give rise to
an increase of
ΔT_{c} = (5/9) ΔT_{f} = (5 / 9) * 9 = 5 degree Celcius
T_{f} = (9 / 5)T_{c} + 32
T_{f} varies linearly as a function of T_{c}
Hence an increase of ΔT_{c} = 10 degree Celcius give rise to
an increase of
ΔT_{f} = (9/5) ΔT_{c} = (9 / 5) * 10 = 18 degree Fahrenheit