MA112 – 1.5 Linear Equations (Solving and Zeros)
Today:
 quiz #4
 1.5: solving linear equations and finding zeros
Announcements
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Homework 1.5 due Mon new setting: HWs can be completed after the due date, but there is a 50% deduction
Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable . a*b +c = d … is a linear equation in a, b, c and d a*b +c
2
= d … is a linear equation in a, b and d BUT NOT c a*x+y=0 … is a linear equation in a, x and y
Solving Equations
Simplify equations by adding or multiplying quantities to both sides.
7 = 7 therefore... 7+2 = 7+2 (9=9) and 7*11=7*11 (77=77)
Example 1:
Solve for x :
5x-4 = 2x+5
5x=2x+9
3x=9 x=3
Example 3a:
Solve for x :
4(5y+3)=2(7-x)
20y+12 = 14-2x
20y-2 = -2x
1-10y = x
Example 2:
Solve for r :
A=P(1+rt)
A=P+Prt
A-P=Prt r=(A-P)/(Pt)
Example 3b:
Solve for y :
4(5y+3)=2(7-x)
20y+12 = 14-2x
20y = 2-2x y = 1/10-x/10

Finding Zeros
“A zero” of a function means the value of the input that yields 0 as an output.
(Since the output is 0, this means y is zero, which means the graph will always be located on the x-axis at a zero.)
A number c is called a zero of the function f if f(c) =0. The graph of y=f(x) crosses the xaxis at x=c precisely when c is a zero of f.
Examples
Find the zeros of f(x). f(x)=x+5 For what inputs (for what values of x) will this function be 0? plug in 1 … get 6 plug in 2 … get 7 plug in -5 … get 0 algebraically, you find the zero by setting the function equal to 0 and solving the equation:
0=x+5 x=-5 … GRAPH to verify that there is a zero at x=-5. f(x) = -3x+13 zeros: 0=-3x+12
3x=12 x=4
You will also need to be able to find the zeros from a graph.

Applications of Finding Zeros
Finding zeros are very useful in applications, as we shall see in subsequent sections.
For example, a zero indicates the possibility that a graph has changed from negative to positive, or from positive to negative. But not necessarily.
Applications of Solving Equations
This is the reason for linear algebra!
Definition of Percent n n % means
100
Example :
34
34% means
100
=0.34
34% of the days were rainy
34
means
100 days were rainy
means 34 out of 100 days were rainy