Overview of Chapter 3
• Slope
• Y=mx+b
• Line of best fit
• Barbie Bungee
• Point-slope equation
• Systems of Equations
 Graphing
 Elimination
 Substitution
Recursive Explicit Linear
Equations
3.1
Goal
• Given one form if a linear equation, convert it
to one of the other forms.
Remember when….?
• What does the graph of an arithmetic
sequence look like?
• We know there is another way calculate linear
equations other than knowing the previous
term right?
• Recursions are ONE type of equation. We will
learn the other EQUIVALENT forms.
Recursive
• 𝑈𝑛 = 𝑈𝑛−1 + 𝑑
• Find the next term by looking at the previous
Explicit
• 𝑈𝑛 = 𝑎 ∙ 𝑛 + 𝑏
• b = Y-intercept. The initial value (𝑈𝑜 ) in the
recursion.
• a= Slope (d in the recursion)
• Nice because you do not have to know the
previous term to find the next.
Linear
• y=mx+b
• m=slope
• b=y-intercept
• Linear uses x and y.
So…
You will be given one of the three types just
discussed, and will be asked to write it in a
different way.
Example 1
• Given the recursion 𝑈0 = 2, 𝑈𝑛 = 𝑈𝑛−1 + 6
1. Find the explicit formula
2. Find 𝑈22 using the explicit
3. Find n such that 𝑈𝑛 = 86
Example 1: answers
1. 𝑈𝑛 = 6𝑛 + 2
slope
initial value
2. 𝑈22 = 6 22 + 2
𝑈22 = 134
3. 86=6n+2
n=14
You try!
• Given the recursion 𝑈0 = 5, 𝑈𝑛 = 𝑈𝑛−1 − 2
1. Find the explicit formula
2. Find 𝑈8 using the explicit
3. Find n such that 𝑈𝑛 = −35
Example 2
• You spend $2 a day on lunch and have $17 left
after today.
Write a recursive and explicit formula modeling this
situation.
So:
Recursive: 𝑈1 = 17 𝑎𝑛𝑑
Explicit: 𝑈𝑛 = −2𝑛 + 17
𝑈𝑛 = 𝑈𝑛−1 − 2
Example 3
• Write an equation in the form y=a +bx of the line
the passes through the points of an arithmetic
sequence with 𝑈0 = 20 and a common
difference of -5.7.
• Answer:
𝑈0 = 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 𝑎 = 20
-5.7=slope=b
y=20-5.7x
Homework
• 3.1
• Problems: 1,4,5