2.3 Linear Equations
We have covered linear functions: e.g. f(x) = 3x + 2.
In order to find the x-intercept of this function, we had to
solve a linear equation 3x + 2 = 0. We now look more
deeply into solving linear equations
solve an equation: find all values of the variable that
satisfy the equation (make it true)
e.g. 3x + 2 = 0; x = -2/3 is the only value that satisfies it
it is not satisfied by, e.g. x = 1
When you solve, one of three things can happen:
1. You obtain a value for the variable (e.g. “x = 10”). The
equation is a conditional equation: satisfied by some but
not all real numbers. Example: 3x - 2 = 0
2. You obtain an equation that is always true; e.g. “0 = 0”.
The equation is identity: satisfied for all values of the
variable. Example: (x + 1)2 = x2 + 2x + 1
Stated solution: “all real numbers”
3. You obtain an equation that is never true; e.g. “1 = 0”
The equation is a contradiction: satisfied for no values.
Example: 2x + 1 = 2x + 3 Stated solution: “no solution”
Word problems
Some HW word problems (marked with a “W”) require
special presentation rules in order to make them readable
and gradable. If you fail to follow these rules, credit will
be withheld.
2.3-1
Presentation rules for word problems (marked “W”)
Please solve these problems as follows:
1. Name and define a variable to be solved for
2. Write an equation (using the variable)
3. Solve the equation
4. State your answer to the question asked
Example: In 2 hours, an athlete travels 20.2 miles by
running first at 8 mph then at 11 mph. How much time did
she run at each speed?
1. name and define the variable (only 1) to be solved for
wrong: let x = runner wrong: let x = speed of runner
wrong: let x = time for runner
wrong: let x = time at 8 mph, y = time at 11 mph
right: let x = time for runner at 8 mph
2. draw a picture, if lengths or distances are involved, and
label appropriately, using the variable, if possible.
8 mph
11 mph
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|
total 20.2 mi
3. write a rough equation, if it will help:
distance@8mph + distance@11mph = 20.2
4. write a finished equation:
using d = rt:
8x + 11(2 - x) = 20.2
5. solve and check (here omitted): x = .6
6. state your answer (do not mention variable):
wrong: x = .6
wrong: time at 8 mph was .6 hr.
wrong: time at 8 mph was .6, at 11 mph . was 1.4
right: time at 8 mph was .6 hr, at 11 mph . was 1.4 hr
2.3-2
Recap: here’s what your solution should look like:
In one hour, and athlete travels 10.1 miles by running first
at 8 mph then at 11 mph. How much time did she run at
each speed?
let x = time for runner at 8 mph
8 mph
(required)
11 mph
|
|
total 20.2 mi
(picture required, if lengths or distances are involved)
distance@8mph
8x
+ distance@11mph = 20.2 (optional)
+ 11(2 - x)
= 20.2 (required)
8x + 22 - 11x = 20.2
3x = 1.8
x = .6
(required)
time (8 mph) = .6 hr, time (11 mph) = 1.4 hr
(required)
2.3-3