Download Linear Equations

Linear Equations
Sections 10.1 – 10.3
►I
can recognize linear
equations.
►I
can solve equations
of the form x + B = C.
►I
can solve equations
of the form Ax = C
►I
can solve equations
of the form Ax + B = C
Write an Equation for the Following
►I
added $30 to my bank account and my
balance is $330. What did I start with?
►I
paid $50 for 10 bags of cherries. How
much did each bag cost?
► Parking
costs a flat rate of $3.00 plus $2.00
per hour. I spent $13.00. How long was I
parked?
These Are Linear Equations
►x
+ 30 = 330
► 10x
► 2x
= 50
+ 3 = 13
► Any
equation that can be written in the form Ax +
B = C, where A, B, C are real numbers.
NOT Linear equations
► x2
+ 5x -3 = 0
► |x
- 3| = 7
► 1/x
= 12
► √x
= 25.
Linear or Nonlinear?
►5
= 2x
►3
–s=¼
►3
– t2 = ¼
► 50
= ¼ r2
Goal: Solve Linear Equations
► We
have simplified expressions with one or
sometimes more than one variable.
► Today
we are going to learn how to solve
linear equations in one variable.
Try This Problem
►I
deposited $30 into my bank account and
my new balance was $330. What did I start
with?
► How
► Try
did you figure this out?
to solve this equation: x + 3 = 7.
This means: what value of x makes the sentence true?
Simplifying Expressions
► When
you simplify expressions, you only
have one side of a scale. The weight
cannot change.
2(x+3)
2x+6
Solving Equations
► When
you solve equations, you have both
sides of the scale! You can change the
weight, but the scale must balance!
x+3=7
x+3-3=7-3
x=4
Inverse Operations
► To
solve x + 3 = 7, you subtract 3 from both sides.
► To
solve x - 3 = 7, what do you do?
► Addition
► We
and subtraction are inverse operations.
always use inverse operations to solve
equations.
More Examples
►5
– k = 12
► 5m
►y
+ 4 = 6m
+ 2/3 = ½
►½
If -x = a, then x = -a
6m – 5m = m (like terms)
Always check your answer!
x – 5 = -1/2 x + 2
Try Some
►x
– 17 = 25
► 12
►t
–r=7
– ½ = 3/4
Don’t forget to check your answer!
Simplify First
► Solve
5t – 4t + 6 = 9
► Simplify first, then solve
5t – 4t = t
t+6=9
► Solve
4x + 6 + 2x – 3 = 9 + 5x – 4
► Solve: 3(2+5x) – (1+14x) = 6
Use the distributive property!
What About Multiplication?
► If
I paid $50 for 10 bags of cherries, how
much did each bag of cherries cost?
► How
did you figure this out?
► Solve:
5x = 60
Try Some More
► Solve
-25p = 50
► Solve
2m = 15
► Solve
-6x = 14.
Fractions I
► In
algebra, instead of writing x ¥ 3, we
write x/3.
► This
is consistent with the notion that
division is multiplication by the reciprocal.
► To
solve x/3 = 10, how do we undo the
division?
Fractions II
► What
► To
about the equation 2/3 x = 6?
divide by 2/3, multiply by the reciprocal.
► Since
(3/2) £ (2/3) = 1, we now have:
x = (3/2) £ 6.
► Work
this out, what do we get?
x=9
Let’s do Another Together
► Solve:
7/5 x = 9/4
► What is the reciprocal of 7/5? 5/7
► Multiply both sides by 5/7.
► What do we have on the left? Just x
The product of a number with its reciprocal is 1!
► For
the right, what is 5/7 times 9/4? 45/28
► What is our solution? x = 45/28
Try These
► Solve
y/12 = 5
► Solve
1/3 z = 19
► Solve
6/7 t = 9/5
Simplify First
► Just
like before, sometimes you have to
simplify before solving!
► Solve:
► Solve
5x + 6x = 9
7x – 2x = -25
Let’s Put These
Together!
► Parking
costs a flat rate of $3.00 plus $2.00
per hour. I spent $13.00. How long was I
parked?
► Answer
► How
the question with your partner.
did you solve this problem?
The Easiest Way To Do This
► The
equation for this problem is:
2h + 3 = 13.
► The variable h stands for hours.
► First:
subtract the 3 from both sides:
2h = 10
► Second: divide both sides by 2:
h = 5.
Another example
► Solve
3x -5 = 7
► First:
addition and subtraction
Add 5 to both sides
► Second:
The order is the opposite!
3x = 12
multiplication and division
Divide both sides by 3
x=4
Try These:
► Solve:
5x – 6 = 17
► Solve:
-4x + 2 = 9
► You
can solve linear
equations using
inverse operations.
► If
you have to deal
with both addition
and multiplication,
deal with addition
first.