Download Solving Linear Equations

Solving Linear Equations
1.5 through 1.7
Equations
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•
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Is 2 + 4 = 6 a true equation?
Is 3 – 5 = 10 a true equation?
Is 2x + 1 = 7 a true equation?
This equation is a conditional equation,
since it depends on x.
Linear Equations
The graph of any linear equation is a straight
line.
Solving Linear Equations:
1. If there are fractions, multiply everything by the LCD.
2. Get rid of ( )’s and combine like terms.
3. Isolate the variable (get it by itself on one side of
equation).
4. Check your answer!
Example 1
Determine whether 3 is a solution of 2x + 4 = 10
Solution
To find out, check the proposed solution, x = 3:
2x + 4 = 10
2(3) + 4 = 10
6 + 4 = 10
10 = 10
This is the original equation.
Substitute 3 for x.
Multiply.
Add.
Since 10 = 10, 3 is a solution, or root, of the equation.
Example 3
Solve 3(x - 2) = 20.
Solution:
3(x - 2) = 20
3x – 6 = 20
3x = 26
26
x=
3
Original equation.
Use the distributive property.
Add 6 to both sides.
Divide both sides by 3.
Example 5
Solve
5
3
( x  3)  32 ( x  2)  2
Solution:
5
3
( x  3)  32 ( x  2)  2
653 ( x  3)  632 ( x  2)  2
6  53 ( x  3)  6  32 ( x  2)  6  2
This is the given equation.
Multiply both sides by 6 (LCD).
Use distributive property.
10(x - 3) = 9(x - 2) + 12
10x - 30 = 9x - 18 + 12
Use distributive property again.
10x – 30 = 9x - 6
Combine like terms.
x = 24
Isolate variable.
Linear Equations
The graph of any linear equation is a straight
line.
Solving Linear Equations:
1. If there are fractions, multiply everything by the LCD.
2. Get rid of ( )’s and combine like terms.
3. Isolate the variable (get it by itself on one side of
equation).
4. Check your answer!
Types of Equations
Conditional:
An equation that is true for at least one
value of “x” (it’s sometimes true).
Identity:
An equation that is true for all values of
“x” (it’s always true).
Contradiction: An equation that has no solution (it’s
never true).
Example 7
Solve 2(x +1) – x = 3(1 + x) – (2x + 1):
2(x +1) – x = 3(1 + x) – (2x + 1)
2x + 2 – x = 3 + 3x – 2x – 1
x+2=x+2
Original equation.
Use the distributive property.
Combine like terms.
Last equation is always true for any value of x
it is an identity!
Example 8
x 1
3
Solve
 4 x  32  13x32
Solution:
x 1
3
 4 x  32  13x32
6 x31   6(4 x)  6( 32 )  613x32 
Original equation.
Multiply everything by 6 (LCD).
2( x  1)  6(4 x)  3(3)  2(13x  2) Simplify fractions.
2x  2  24x  9  26x  4
26x – 2 = 26x + 5
-2 = 5
Use distributive property.
Combine like terms.
Subtract 26x from both sides.
Since –2 = 5 is never true, there is no solution
it is a contradiction!
Time Trial
Determine whether the equation 3(x - 1) = 2(x + 3) + x
is an identity, a conditional equation, or a contradiction.
Solution
To find out, solve the equation.
3(x – 1) = 2(x + 3) + x
3x – 3 = 2x + 6 + x
3x – 3 = 3x + 6
-3 ≠ 6
This equation is a contradiction.
Formulas
A formula is an equation involving two or
more variables.
D = RT
A = LW
P = 2L + 2W
Example 10
Solve for t in the following formula: A = p + prt.
Solution: To solve for t means to isolate it on on side.
A = p + prt
Original equation.
A – p = prt
To isolate t, subtract p from both sides.
A p
pr
t
Divide both sides by pr.
t
A p
pr
Write t on left-hand side.