Transcript Equations with Two Variables Notes

Constant Rate of
Change
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Lesson 1
Marcus can download two songs from the Internet each
minute. This is shown by the table below.
a. Compare the change in number of songs y to the change
in time x. What is the rate of change?
The number of songs increases by 2, and the time
increases by 1.
b. Graph the ordered pairs.
Describe the pattern.
The points appear to make
a straight line.
Real World Link
1. Relationships that have a straight-line graph
2. The rate of change is the same – “constant rate of
change”
Linear Relationships

The balance in an account after several transactions is
shown. Is the relationship linear? If so, find the constant
rate of change.
Since the rate is constant, this shows a linear relationship.
The constant rate of change is
involved a $10 withdrawal.
Example 1
−30
3
or -10. Each transaction
a.
Got it? 1
b.
Words: Two quantities a and b have a proportional linear
relationship if they have a constant ratio and a constant rate
of change.
Symbols:
𝑏
𝑎
is constant and
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑏
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎
is also constant.
2
4
6
8
Example: there is a constant rate of change and = = =
1
2
3
4
Proportional Linear Relationships

Use the table to determine if there is a proportional linear
relationship between the temperature in degrees and Fahrenheit and
a temperature in degrees Celsius. Explain your reasoning.
41
50
= 8.2,
= 5,
5
10
Since the ratios are not
equal, degrees Fahrenheit
and Celsius are not
proportional.
Since the rate of change is
constant, this is a linear
relationship.
Example 2

Use the table to determine if there is proportional linear
relationship between the mass of an object in kilograms and
the weight of the object in pounds. Explain your reasoning.
Weight (Ibs)
Mass (kg)
20
9
40
18
60
27
80
36
Got it? 2
Slope
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Lesson 2
Slope =
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛

Find the slope of the treadmill.
Slope =
=
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
10 𝑖𝑛𝑐ℎ𝑒𝑠
48 𝑖𝑛𝑐ℎ𝑒𝑠
=
5
24
The slope of the treadmill is
5
.
24
Example 1
A hiking trail rises 6 feet for every horizontal change of 100
feet. What is the slope of the hiking trail?
Slope =
Got it? 1
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙

The graph shows the cost of muffins at a bake sale. Find
the slope of the line.
The vertical change
is 2 and the
horizontal change is
1.
The slope is
2
1
or 2.
Example 2

The table shows the number of pages Garrett has left to
read after a certain number of minutes. The points lie on a
line. Find the slope of the line.
Slope =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
Choose any points: (1, 12) and (3, 9)
Slope =
12 −9
1 −3
=
The slope is -
Example 3
3
−2
3
.
2
a.
Got it? 2 & 3
b.
Symbols:
m = slope
m=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
Slope Formula
Find the slope of the line that passes through
𝑦2 −𝑦1
R(1, 2), S(-4, 3).
m=
𝑥2 −𝑥1
m=
3 −2
−4 −1
m=
1
−5
The slope of the line is -
Example 4
1
.
5
a. A(2, 2), B(5, 3)
Got it? 4
b. J(-7, -4), K(-3, -2)
Equations in y = mx Form
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Lesson 3
Find the slope of this graph.
m=
𝑦 −0
𝑥 −0
m=
𝑦
𝑥
Solve for y.
y = mx
Words: “y varies directly with x”
Symbols: m =
𝑦
𝑥
or y = mx, where m is the constant or slope.
Example: y = 3x
Graph:
Direct Variation
The amount of money Robin earns while babysitting varies
directly with the time as shown in the graph. Determine the
amount that Robin earns per hour.
Find “nice” points on the graph
to find the constant of
variation.
(2, 15) and (4, 30)
m=
m=
15
2
= 7.5
𝑦
𝑥
m=
30
4
= 7.5
So Robin earns $7.5 and hour
babysitting.
Example 1
Two minutes after a skydiver opens his parachute, he has
descended 1,900 feet. After 5 minutes, he descends 4,750
feet. If the distance varies directly with the time, at what
rate is the skydiver descending?
Hint: Find the two points on the line.
Got it? 1
A cyclist can ride 3 miles in 0.25 hour. Assume that the
distance biked in miles varies directly with time in hours x.
This situation can be represented by y = 12x. Graph the
equation. How far can the cyclist ride per hour? 12 miles per hour
Make a table of values.
Example 2
Graph the values.
The slope is 12 since the
equation is y = 12x.
A grocery store sells 6 oranges for $2. Assume that the cost
of the oranges varies directly with the number of oranges.
1
This situation can be represented by y = x. Graph the
3
equation. What is the cost per orange?
Got it? 2
In a proportional relationship, how is the unit rate
represented on a graph?
It is the slope.
When comparing two different direct variation equations,
what’s the difference?
y = 3x
y = 12x
The rate or slope.
Comparing Direct Variation
The distance d in miles covered by a rabbit in t hours can be
represented by d = 35t. The distance covered by a grizzly
bear is shown on the graph. Which animal is faster, or has
the fastest rate?
Rabbit:
Grizzly Bear:
d = 35t, so the
rate is 35.
Find the slope
of the line.
The rate is 30.
35 is greater than 30, so the
rabbit is faster.
Example 3
Damon’s earnings for four weeks from a part time job are
shown in the table. Assume that his earnings vary directly
with the number of hours worked.
He can take a job that will pay him $7.35 per hour. What job
is the better pay? Explain.
Got it? 3
A 3-year old dog is often considered to be 21 in human years.
Assume that the equivalent age in human years y varies
directly with its age as a dog x. Write and solve a direct
variation equation to find the human-year age of a dog that
is 6 years old.
We know that when y is 21, x is 3.
y = mx
21 = m(3)
m=7
The rate is 7.
Example 4
y = 7x
y = 7(6)
y = 42
A dog that is 6 years old has an
equivalent human age of 42.
a.
b.
A charter bus travels 210 miles in 3.5 hours. Assume the
distance traveled is directly proportional to the time
traveled. Write and solve a direct variation equation to
find how far the bus will travel in 6 hours.
A Monarch butterfly can fly 93 miles in 15 hours. Assume
the distance traveled is directly proportional to the time
traveled. Write and solve a direct variation equation to
find how far the Monarch butterfly will travel in 24 hours.
Got it? 4
Slope-Intercept Form
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Lesson 4
Slope-Intercept Form
State the slope and the y-intercept of the graph of the
2
equation y = x – 4.
3
y = mx + b, where m is slope and b is y-intercept
y=
m=
The slope is
Example 1
2
3
2
3
2
x
3
–4
and b = -4
and the y-intercept is -4.
Find the slope and y-intercept for each equation.
a.
y = -5x + 3
1
x
4
b.
y=
–6
c.
y = -x + 5
Got it? 1
Write an equation of a line in slope-intercept form if you
know that the slope is -3 and the y-intercept is -4.
y = mx + b
y = -3x + (-4) or y = -3x – 4
Write an equation of a line in slope-intercept form from the
graph below.
y = mx + b
the slope is 2 and the
y-intercept is 4
1
y=- x+4
2
Examples 2 & 3
a. Write an equation in slope- b. Write an equation in
intercept form for the graph. slope-intercept form with a
3
slope of and a y-intercept
4
of -3.
Got it? 2 & 3
Student Council is selling T-shirts during spirit week. It costs
$20 for the design and $5 to print each shirt. The cost y to
print x shirts is given by y = 5x + 20. Graph this equation
using the slope and y-intercept.
Step 1: Find the slope and y-intercept.
slope = 5
y-intercept = 20
Step 2: Graph the y-intercept (0, 20)
Step 3: Go up 5 and over 1 to find
another point.
Example 4
Student Council is selling T-shirts during spirit week. It costs
$20 for the design and $5 to print each shirt. The cost y to
print x shirts is given by y = 5x + 20. Interpret the slope and
y-intercept.
The slope represents the cost of each T-shirt. The
y-intercept is the one time fee of $20 for the design.
Example 5
A taxi fare y can be determined by the equation y = 0.50x +
3.5, where x is the number of miles traveled.
a.
Graph this equation.
b.
Interpret the slope and y-intercept.
Got it? 4 & 5
Graph a Line Using
Intercepts
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Lesson 5
x and y intercepts
State the x- and y-intercepts of y = 1.5x – 9. Then use the
intercepts to graph the equation.
STEP 1: Find the y-intercept. STEP 3: Graph the two
y-intercept is -9.
intercepts and connect to make
a line.
STEP 2: Find the x-intercept,
let y = 0.
0 = 1.5x – 9
9 = 1.5x
6=x
Example 1
Graph these equations by using the x- and y- intercepts.
1
3
a. y = − x + 5
Got it? 1
3
2
b. y = − x + 3
Standard form is when the equation is Ax + By = C, and A, B,
and C are integers. A, B, and C can NOT be fractions or
decimals.
Take the equation:
A = 60
60x + 15y = 4,740
B = 15
Standard Form
C = 4,740
Mauldlin Middle School wants to make $4,740 from yearbooks.
Print yearbooks x cost $60 and digital yearbooks y cost $15. This
can be represented by the equation 60x + 15y = 4,740.
Use the x- and y-intercepts to graph the equation.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0
60x + 15y = 4740
60x + 15(0) = 4740
60x = 4740
x = 79
Example 2
60x + 15y = 4740
60(0) + 15y = 4740
15y = 4740
y = 316
Graph the equation, using the intercepts, and interpret the
x- and y- intercepts.
The x-intercept is at the point
(79,0). This means they can
sell 79 print yearbooks and
still earn $4,740.
The y-intercept is at the point
(0, 316). This means they can
sell 316 digital yearbooks and
still earn $4,740.
Example 3
Mr. Davis spent $230 on lunch for his class. Sandwiches x
cost $6 and drinks y cost $2. This can be represented by the
equation 6x + 2y = 230. Graph and interpret the x- and yintercepts.
Got it? 2 & 3
Write Linear Functions
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Lesson 6
Point-Slope Form
Write an equation in point-slope form for the line that passes
through (-2, 3) with a slope of 4.
y – y1 = m(x – x1)
y – 3 = 4(x – (-2))
y – 3 = 4(x + 2)
Example 1
Write an equation in slope-intercept form for the line that
passes through (-2, 3) with a slope of 4.
y = mx + b
3 = 4(-2) + b
3 = -8 + b
11 = b
y = 4x + 11
Example 2
Write an equation in point-slope form and slope-intercept
1
form for a line that has a slope of - and a point (-1, 2).
2
Point-Slope Form:
y – y1 = m(x – x1)
Got it? 1 & 2
Slope-Intercept Form:
y = mx + b
STEP 1: Find the slope.
STEP 2: Use one of the points and the slope to make an
equation in point-slope form.
STEP 3: Use one of the points and slope to make an equation
in slope-intercept form.
STEP 4: Find the y-intercept.
STEP 5: Rewrite the slope-intercept problem with slope and
y-intercept.
Writing a Linear Equation
from Two Points
Write an equation in point-slope form and slope-intercept
form for a line that passes through (8, 1) and (-2, 9).
Point-Slope Form:
9 −1
8
4
Slope =
=
=−
Slope-Intercept Form:
y = mx + b
Use the point (8, 1)
y=− x+b
−2 −8
−10
4
5
y – 1 = − (x – 8)
5
4
5
Use the point (8, 1)
4
5
1 = − (8) + b
b=
Example 3
37
5
Write an equation in point-slope form and slope-intercept
form for a line that passes through (3, 0) and (6, -3).
Point-Slope Form:
Got it? 3
Slope-Intercept Form:
The cost of assistance dog training sessions is shown in the
table. Write an equation in point-slope form to represent
the cost y of attending x dog training sessions.
Find the slope.
m=
290 −165
10 −5
=
125
5
= 25
Use the point (5, 165).
y – 165 = 25(x – 5)
y – 290 = 25(x – 10)
Example 4
The cost for making spirit buttons is shown in the table.
Write an equation in point-slope form to represent the cost
of y of making x buttons.
Got it? 4
Solve Systems of
Equation by Graphing
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Lesson 7
System of Equations:
Two or more equations with the same set of variables.
The solution to a system is when they lines cross at a certain
point.
The solution for this system
of equations is (3, 8).
System of Equations
Solve the system y = -2x – 3 and y = 2x + 5 by graphing.
Graph each line on the same coordinate plane.
The solution is (-2, 1)
y = -2x – 3
1 = -2(-2) – 3
1=1
y = 2x + 5
1 = 2(-2) + 5
1=1
Example 1
Graph to find the solution.
y=x–1
y = 2x – 2
Got it? 1
Gregory’s Motorsports has motorcycles (two wheels) and
ATV’s (four wheels) in stock. The store has a total of 45
vehicles, that together, have 130 wheels.
Write a system of equations that represent this situation.
Let x be the # of motorcycles and y be the # of ATV’s.
x + y = 45
and
2x + 4y = 130
Example 2
Gregory’s Motorsports has motorcycles (two wheels) and
ATV’s (four wheels) in stock. The store has a total of 45
vehicles, that together, have 130 wheels.
Solve the system of equations. Interpret the solution.
x + y = 45
2x + 4y = 130
Graph the equations.
The store has 20 motorcycles
and 25 ATV’s.
Example 3
Creative Crafts gives scrapbooking lesson for $15 per hour
plus a $10 supply charge. Scrapbooks Incorporated gives
lessons for $20 per hour with no additional charges. Write
and solve a system of equations that represents the
situation. Interpret the situation.
Got it? 2 & 3
If the lines intersect, there is one solution.
If the lines are parallel, there are no
solutions.
If the lines are the same, there are infinitely
many solutions.
Number of Solutions
Solve the system by graphing.
y = 2x + 1
y = 2x – 3
Since the lines are parallel,
there is no solution.
Example 4
Solve the system by graphing.
y = 2x + 1
y - 3 = 2x – 2
Since the lines are the same,
there are infinitely many
solutions.
Example 5
Solve each system of equations by graphing.
2
x
3
a. y =
+3
3y = 2x + 12
Got it? 4 & 5
b. y – x = 1
y=x–2+3

A system of equations consists of two lines. One line
passes through (2, 3) and (0, 5). The other line passes
through (1, 1) and (0, -1). Determine if the system has one
solution, no solution, or infinite number of solutions.
Carefully graph the points and make the two lines.
The lines appear to cross at (2, 3).
(2, 3) and (0, 5)
Slope = -1
Equation:
y = -1x + 5
(1, 1) and (0, -1)
Slope = 2
Equation:
y = 2x – 1
3 = -2 + 5
3=3
3 = 2(2) – 1
3=3
Example 6

A system of equations consists of two lines. One line
passes through (0, 2) and (1, 4). The other line passes
through (0, -1) and (1, 1). Determine if the system has one
solution, no solution, or infinite number of solutions.
Got it? 6
Solve Systems of
Equations Algebraically
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Lesson 8
Solve the system of equations algebraically.
y=x–3
y = 2x
Since y = 2x, then you can substitute 2x in for the first
equation.
y=x–3
2x = x – 3
x = -3
When x = -3, then y is -6.
The solution is (-3, -6)
Example 1
Solve each system of equations algebraically.
a. y = x + 4
y=2
Got it? 1
b. y = x - 6
y = 3x

Solve the system of equations algebraically.
y = 3x + 8
8x + 4y = 12
8x + 4(3x + 8) = 12
Use the Distributive Property
8x + 12x + 32 = 12
20x + 32 =12
20x = -20
x = -1
x = -1
y = 3(-1) + 8
y = -3 + 8
y=5
The solution is (-1, 5).
Example 2
Solve each system of equations algebraically.
a. y = 2x + 1
3x + 4y = 26
Got it? 2
b. 2x + 5y = 44
y = 6x – 4
A total of 75 cookies and cakes were donated for a bake sale
to raise money for the football team. There were four times
as many cookies donated as cakes.
Write a system of equations to represent this situation.
y = 4x
x + y = 75
Example 3
Solve the system of equations from Example 3 algebraically.
x + y = 75
y = 4x
x + 4x = 75
5x = 75
x = 15
When x is 15, y is 60. The solution is (15, 60). This means
that 15 cakes and 60 cookies were donated to the bake sale.
Example 4
Mr. Thomas cooked 45 hamburgers and hot dogs at a cookout.
He cooked twice as much hot dogs than hamburgers.
a. Write a system of equations that represents this situation.
b. Solve the system algebraically and interpret the solution.
Got it? 3 & 4