Thinking
Algebraically
Equation of a Line
Have you ever…?
• Tested your speedometer by driving a constant speed
between mileposts?
• This situation involves a linear relationship between variables.
For example, when one variable (such as time) increases, the
other variable (such as distance) either increases or decreases
steadily.
• A graph is a good way to visualize this relationship.
A graph is a good way to
visualize this relationship:
Linear Equations on a Graph
• In a linear relationship, the rate of change, is constant for each
variable. For example, if a car travels at a constant speed, it’s
distance traveled will be constant also.
• On a graph, the slope of the line is constant, too. It doesn’t
change.
Points on a Grid
•
•
•
•
Find these points on your grid:
(x, y)
(2, -3)
(4, -1)
Slope-Intercept Form
• Slope Intercept form
is the easiest way to
graph a linear
equation.
“m" is the slope (or
the steepness of the
graph)
“b" is the y-intercept
(where the graph
crosses the y-axis).
Graphing a line when slope and
y-intercept are known
• To graph using
y=mx +b, start by
putting a point on the yintercept.
Then, move depending
on the
slope. Remember that
slope is rise/run. So
move up if the slope is
positive and over to the
right. If the slope is
negative move down
and over to the right.
• y = 3x +12
Slope Formula
• How to find the slope of a line that contains two points (x,y) and
(x,y)
1. Label each coordinate (x1, y1) and (x2, y2)
2. Substitute them into the equation form
3. Evaluate
Example: Find the slope of the line through (2,4) and (-5, -7)
(2, 4) (-5, -7)
(2 , 4) (-5, -7)
x2, y2
x1, y1
x1, y1
x2, y2
m = (4 - -7) = (4 + 7) = 1 1
(2 - -5) (2 + 5) 7
Slope is 11/7
Mistakes to Watch out for
when finding slope
• Three common mistakes are
• Forgetting that y goes on top.
• Switch the order of the points between top and bottom.
• Make a mistake with negatives.
Sample Problem
• Your company’s productivity (x) has a linear relationship with
complaints (y). When productivity is three below the target,
there are six complaints above the minimum goal (-3, 6).
When productivity is three above the target, there are two
complaints less than the minimum goal (3, -2). Write a linear
equation that represents the relationship of productivity to
complaints in slope-intercept form.
Find the slope
• Calculate the slope m of a line by finding the “rise over run.”
• Use this formula:
• Use these points from our sample problem: (-3, 6)(3, -2).
•
( x1,y1)(x2,y2)
• M = change in y = -2 - 6 = -8 = -4
•
change in x = 3 – (-3)= 6 = 3
Choose the form
• Since you know the slope and two points, you can use the
point-slope form.
• slope-intercept or point-slope form?
Graphing a line when the slope
and one point is known.
• y-(-2)= -4(x-3)
•
3
•
-4 (x) + -4 * -3
•
3
3 1
• Y +2 = -4 x +4
•
3
• -2
-2
• Y= -4 x +2
•
3
p. 136
• Niveah is concerned that her speedometer is not reading
properly. On the highway, she decides to test it. While holding
the car steady at 60 miles per hour according to her
speedometer, she passes mile marker 230 and starts the
stopwatch on her phone. She passes mile marker 235 exactly
4.8 minutes later.
Thinking
Algebraically
Graphing Two-Variable Equations
Have you ever…?
• Played Battleship
• The game “Battleship” uses a coordinate grid, just like we do
in math.
Some terms and definitions:
• Axes (sing. axis): lines that represent the things whose
relationship is being compared.
• Coordinate plane (coordinate axes): When two lines/axes
intersect at a right angle, creating a grid.
• Absolute value: the distance of a number from zero on the
number line.
Positive or negative slope?
(or zero slope or undefined slope)?
Slope-intercept form
• A line can be described in slope-intercept form with its slope,
m, and its y-intercept, b:
•
y=mx +b
• This equation gives you a lot of information about the line:
•
•
•
•
If the slope is positive, the line climbs from left to right.
If the slope is negative, the line falls from left to right.
The larger the absolute value of the slope, the steeper the line.
The equation shows you the y-intercept, where x= 0
One, Two, Three, Graph!
• Graphing is a way to represent data visually. An equation like
y=7x+3 tells you a lot about the relationship between x and y.
• But the equation can’t instantly show you what y is if x is 23. A
graph can. You can draw a graph in three steps:
• Find three points that satisfy the equation (make the equation
true).
• Draw the points on the coordinate plane.
• Draw a straight line through the three points.
• The cost to rent a copier for a day is $50 plus $0.05 per copy.
To help her employees understand the company’s expenses,
Chanry posts a graph of the total costs per copy of the copier.
What does the graph look like?
Find the equation to use
• Use the slope-intercept form y=mx+b or the point-slope form y-y1= m(xx1) to find an equation for the graph. The slope (m) is the rate of
change, and the y-intercept is a constant value.
• Use the point-slope form if you know a point on the graph and the
slope.
• Use the slope-intercept form if you know the slope and the y-intercept.
• The cost to rent a copier for a day is $50 plus $0.05 per copy. To help
her employees understand the company’s expenses, Chanry posts a
graph of the total costs per copy of the copier. What does the graph
look like?
• Which equation should we use?
• Use the slope-intercept form, since you have a rate of change and a
constant value. The rate per copy is 0.05, so it is the slope (m). The yintercept (b) is the amount of y if x = 0. In this case, it is the cost per day
if the office makes zero copies ($50). The equation is:
• Y= 0.05x + 50
Why make a table of x/y
values?
• You could graph the line using the slope and y-intercept, but if
you graph the line using three points, you can check for errors.
Y= 0.05x + 50
• Pick three easy to use values of x, such as 0, 1, and 2 or 0, 100,
and 200
X
Y
0
50
100
55
200
60
What does this chart tell you?
(Based on this chart, would you change your copier situation?)
Cost of Copies
70
60
50
40
30
20
10
0
Y-Values
0
50
Number of Copies
100
150
200
250
Graph this linear equation:
y= 7x + 3
Y-Values
12
10
X
Y
0
3
1
10
-1
-4
8
6
4
Y-Values
2
0
-3
-2
-4
-6
2
7
12
2y+6=x+10
•
•
•
•
•
2y+6-6=x+10-6
2y=x+4
2y=x+4
2
2
y= 1/2x +2
• Y= ½(0)+2
• y=0+2
• y=2
• y=1/2 (2)+2
• y=1+2
• y=3
Thinking
Algebraically
Systems of Equations
Have you ever…?
• Bought adult and child tickets to a movie?
• Made a budget when costs of two utilities varied month to
month?
• In the real world, you often have more than one unknown
value. Whenever you can draw relationships between two
values in two different ways (for example, the amount charged
on electric and gas in May and then in June), you can find two
unknown values using a system of equations.
New Vocabulary:
• A system of equations is a group of related equations that
deal with the same variables. If you have the same number of
equations as unknown values, you can find a single solution, if
there is one.
In some cases, a group of
equations can’t all be true.
• For example: x + y = 5 and x + y = 10
• These can’t both be true if x and y represented the same value
in both equations.
• Or, take a look at these two equivalent equations:
• X +y = 2 and x= 2-y
• This is the same equation, just reorganized.
• However, if a system of different equations correctly represents a
real-world situation, you can use them to find the unknown
values.
Solve by substitution
• Imagine you have two equations, 2x = y and y = x + 1. To solve
a system of two equations by substitution:
• Put one equation in the form
• variable = expression.
• You could use y = x + 1
• or change 2x = y to x = y/2
• Substitute the expression for the variable in the other equation.
For example, substitute x + 1 for y in 2x = y to get 2x = x + 1
• Solve the equation. When you solve 2x = x + 1, you get x = 1.
• Use the solution to find the other variable. If x = 1, and y = x + 1,
then y = 2.
• Tobin is considering two investment opportunities. Plan A
offers 4% interest, and Plan B offers 8% interest. He has
$5,250 to invest and is wary of the risks associated with the
higher interest rate. He decides to invest three times as much
in Plan A as in Plan B. How much does he invest in each plan?
• Tobin is considering two investment opportunities. Plan A
offers 4% interest, and Plan B offers 8% interest. He has
$5,250 to invest and is wary of the risks associated with the
higher interest rate. He decides to invest three times as much
in Plan A as in Plan B. How much does he invest in each plan?
• How much he invests in plan A: a
• How much he invests in plan B: b
• a + b = 5250
• a = 3b
a + b = 5250 and a = 3b
• Examine your equations. Does one of them have one variable
isolated on one side?
• Yes. a = 3b
• Substitute the equivalent expression for that variable in the
other equation and simplify.
•
•
•
•
a + b = 5250
3b + b = 5250
4b = 5250
b = 1,312.50
Solve for the other variable
• Now you know one of the variables, but you need a value for
the other variables. Substitute the value you found into either
of the equations from your system of equations and simplify.
a + b = 5250
a + (1312.5) = 5250
-1312.5 -1312.5
a
= 3937.5
a = 3b
a = 3 (1312.5)
a = 3937.5
p. 152
• 2x + 7y = 4
• y = -x – 3
•
•
•
•
•
•
•
•
2x + 7(-x – 3) = 4
2x – 7x -21 = 4
-5x – 21 = 4
-5x -21 + 21 = 4 + 21
-5x = 25
-5x = 25
-5 -5
x = -5
• y = -(-5) -3
• y = +5 – 3
• y=2
Is there a different method to
solve this system of equations?
• Solve for x in the
second equation.
• y = -x – 3
• y + 3 = -x – 3 + 3
• y + 3 = -x
• y + 3 = -x
• -1
-1
• -y – 3 = x
• Substitute for x in the
first equation.
• 2(-y – 3) + 7y = 4
• -2y – 6 + 7y = 4
• 5y -6 = 4
• 5y -6 +6 = 4 + 6
• 5y = 10
•Y=2
• A community theater produced a performance of Mama Mia.
It sold regular tickets for $18 and senior and student tickets
for $10. On Friday, it brought in $2,474. It sold 85 more
discount tickets than regular tickets. How many tickets of each
type were sold?
• A community theater produced a performance of Mama Mia.
It sold regular tickets for $18 and senior and student tickets
for $10. On Friday, it brought in $2,474. It sold 85 more
discount tickets than regular tickets. How many tickets of each
type were sold?
• s = number of senior/student tickets sold
• r = number of regular tickets sold
• r +85 = s
• 18r + 10s = 2474
• r +85 = s
• 18r + 10s = 2474
•
•
•
•
•
•
18r + 10(r + 85) = 2474
18r +10r + 850 = 2474
28r + 850 = 2474
-850
-850
28r = 1624
r= 58
• r + 85 = s
• 58 + 85 = s
• 143 = s
Solve By Addition p. 154
• We just learned how to solve equations by substitution. Now,
I’m going to teach you another way to solve a system of
equations: by addition. This is similar to adding numbers by
lining up place values.
• 1255.54
• +879.11
• 2134.65
Solve by addition
1. Find opposite coefficients (positive and negative numbers
with the same absolute value: such as (-5) and 5).
2. Write the equations so that like terms are in the same order.
3. Add the equations term by term, eliminating the variable
with opposite coefficients.
4. Solve and use the solution to find the other variable.
• A motorboat travelling with the current went 24 miles in two
hours. On the return trip, it took 2.5 hours to travel the same
distance. Find the rate of the boat and of the current.
• A motorboat travelling with the current went 24 miles in two
hours. On the return trip, it took 2.5 hours to travel the same
distance. Find the rate of the boat (b) and of the current(c).
• b is the speed of the boat, and c is the speed of the current.
• When the boat is travelling with the current, the speed is
added to the current. b+c= 12 (24/2)
• When the boat is travelling against the current, the rate of the
current is subtracted from the rate of the boat. b-c = 10
The Steps
1. Find opposite coefficients (positive and negative numbers
with the same absolute value: such as (-5) and 5).
2. Write the equations so that like terms are in the same order.
3. Add the equations term by term, eliminating the variable
with opposite coefficients.
4. Solve and use the solution to find the other variable.
b+c=12 and b-c= 10
• The key to adding equations
is to eliminate one of the
variables.
Substitute our value for b back
into the original equation and
solve for c.
•
•
•
•
•
•
•
• b + c = 12
• 11+c= 12
• -11 -11
•
c=1
b +1c = 12
b –1c = 10
2b+0 = 22
2b = 22
2b =22
2
2
b = 11
• The boat is going 11mph and the current is going 1mph.
Thinking
Algebraically
Functions
Have you ever…?
• Used military time such as 0800 hours?
• Converted inches to centimeters?
• Determined profit or loss of a business venture or fundraising
activity?
These are all examples of relations. A relation is a relationship
between two variables that creates ordered pairs (x,y). Any
equation with x and y is a relation. A function is a special
relation that follows an additional rule. For every x, there can
only be one y.
A function is like a machine.
• You put in a value (x), such as five inches, and it gives you a
single corresponding value f(x), such as 12.7 centimeters.
• Functions can be very useful. For example, the time 12:00 can
be very confusing. The number 12 can point to two values,
noon and midnight. However, military time goes from 00
hundred hours to 2400 hours. 1200 hours means noon, not
midnight.
• Military time is a function: one number means exactly one
time.
• The domain: the set of input values (x).
• The range: is the set of output values that the
function(machine) gives you for x, graphed on the y-axis.
The domain and range
For every value of x, there is
only one value for y.
Understanding functions
• f(x) is the same as y; therefore, it refers to the value
associated with a given x.
• y= x+2
or
f(x) = x+2
• f(1) means, “Evaluate the equation (find y) when x= 1.”
Another way to ask this is, “What output do you get for this
input?”
• In our example equation above, the f(1) is 3:
• f(x) = x+2
• f(1) = 1 + 2
• f(1) = 3
Is this a function or not?
The Vertical Line test.
The Value of a Function at a
Point
• To find the f(x) using a graph, simply locate the y value for the
specific x.
• Find f(1) and f(0) on graphs A and D.
Rise over run
• use the slope formula
Describing Graphs of Functions
• Intercepts: This is where the line crosses either the x or y axis.
• Maximum or minimum: The highest point on the graph is the
maximum, and the lowest point is the minimum.
• Positive or negative: When y is positive, the function is
positive. When y is negative, the function is negative.
• Increasing or decreasing: When the slope is slanting upwards,
the function is increasing. When the slope is slanting
downwards, the function is decreasing.