Linear Equations and Rational Equations
Section 1.1
Solve Linear Equations in One Variable
Solving a Linear Equation in One Variable
Step 1
Simplify both sides of the equation.
•
•
•
Use the distributive property to clear parentheses.
Combine like terms.
Consider clearing fractions or decimals by multiplying both sides of the equation by the least common
denominator (LCD) of all terms.
Step 2
Use the addition property of equality to collect the variable terms on one side of the equation
and the constant terms on the other side.
Step 3
Use the multiplication property of equality to make the coefficient on the variable term equal to 1.
Step 4
Check the potential solution in the original equation.
Step 5
Write the solution set.
1.
Solve: x + 4 ( x − 5) = 4 ( x − 2)
2.
Solve: 2x − 3( 2x + 2) = 1 − 5 ( 4x + 3)
3.
Solve: 0.6x −1.3 = 0.2 ( 5x − 8)
4.
Solve:
2
5
1
x − x −3= x −5
3
6
2
Identify Conditional Equations, Identities, and Contradictions
A conditional equation is true for some values of the variable and false for other values.
An identity is an equation that is true for all values of the variable for which the expressions in an equation are
defined.
A contradiction is an equation that is false for all values of the variable.
5. Solve: 24 + 2(5 − x ) − 2 x  = 4(7 − 2 x )
7.
Solve: − 3( x − 2) + 4(2 x + 5) = 10 x − (5x − 23 )
6.
Solve: 2 4 + 2 ( 5 − x ) − 2 x  = 4 ( 7 + 2 x )
Solve Rational Equations
A rational equation is an equation in which each term contains a rational expression. All linear equations are
rational equations, but not all rational equations are linear. When a variable appears in the denominator of a
fraction, we must restrict the values of the variable to avoid division by zero.
8.
Solve:
9 1
4
−
=
4 2x x
restricted value for x : __________
9.
Solve:
x+6
3
−2=
x+3
x+3
restricted value for x : __________
10. Solve:
1
y +1
1
+ 2
=
y −1 y + 2y − 3 y + 3
restricted values for y : ____________
Solve Literal Equations for a Specified Variable
11. 10x − 2a = 3a
Solve for x.
3
( z + 1)
2
Solve for z.
13.
A=
12. 3x − 5 y = 10
Solve for y.
14. 3x + 4a = 2 px + 1
Solve for x.
15. Adele's 2012 Mini-Cooper gets 37 miles per gallon on the highway and 29 miles per gallon in the city. The
amount of gas she uses A (in gallons) is given by
A=
1
1
c+ h
29
37
where c is the number of city miles driven and h is the number of highway miles driven.
If Adele drove 58 miles in the city and used 5 gallons of gas, how many highway miles did she drive?