2-4 Solving Equations With Variables on Both Sides

Quick Review

When an equation has variables on both sides, you can use properties of equality to isolate the variable on one side. An equation has no solution if no value of the variable makes it true. An equation is an identity if every value of the variable makes it true.

Example

What is the solution of 3 x − 7 = 5 x + 19 ? 3 bold italic x minus 7 equals 5 bold italic x plus 19 question mark

3 x − 7 − 3 x = 5 x + 19 − 3 x Subtract  3 x . − 7 = 2 x + 19 Simplify . − 7 − 19 = 2 x + 19 − 19 Subtract  19 . − 26 = 2 x Simplify . − 26 2 = 2 x 2 Divide each side by  2 . − 13 = x Simplify . table with 6 rows and 3 columns , row1 column 1 , 3 x minus 7 minus 3 x , column 2 equals 5 x plus 19 minus 3 x , column 3 cap subtract . 3 x . , row2 column 1 , negative 7 , column 2 equals 2 x plus 19 , column 3 cap simplify , . , row3 column 1 , negative 7 minus 19 , column 2 equals 2 x plus 19 minus 19 , column 3 cap subtract . 19 . , row4 column 1 , negative 26 , column 2 equals 2 x , column 3 cap simplify , . , row5 column 1 , negative 26 over 2 , column 2 equals , fraction 2 x , over 2 end fraction , column 3 cap divideeachsideby . 2 . , row6 column 1 , negative 13 , column 2 equals x , column 3 cap simplify , . , end table

Exercises

Solve each equation. If the equation is an identity, write identity. If it has no solution, write no solution.

24. 2 3 x + 4 = 3 5 x − 2 2 thirds , x plus 4 equals , 3 fifths , x minus 2
25. 6 − 0.25 f = f − 3 6 minus , 0.25 , f equals f minus 3
26. 3 ( h − 4 ) = − 1 2 ( 24 − 6 h ) 3 . open , h minus 4 , close . equals negative , 1 half . open . 24 minus 6 h . close
27. 5n = 20(4 + 0.25n)
28. Architecture Two buildings have the same total height. One building has 8 floors with height h. The other building has a ground floor of 16 ft and 6 other floors with height h. Write and solve an equation to find the height h of these floors.
29. Travel A train makes a trip at 65 mi/h. A plane traveling 130 mi/h makes the same trip in 3 fewer hours. Write and solve an equation to find the distance of the trip.

2-5 Literal Equations and Formulas

Quick Review

A literal equation is an equation that involves two or more variables. A formula is an equation that states a relationship among quantities. You can use properties of equality to solve a literal equation for one variable in terms of others.

Example

What is the width of a rectangle with area 91  ft 2 91 , ft squared  and length 7 ft?

A = l w Write the appropriate formula . A l = w Divide each side by  l . 91 7 = w Substitute  91  for  A  and  7  for  l . 13 = w Simplify . table with 4 rows and 3 columns , row1 column 1 , eh , column 2 equals l w , column 3 cap writetheappropriateformula . . , row2 column 1 , eh over l , column 2 equals w , column 3 cap divideeachsideby . l . , row3 column 1 , 91 over 7 , column 2 equals w , column 3 cap substitute . 91 , for eh , and , 7 , for , l . , row4 column 1 , 13 , column 2 equals w , column 3 cap simplify , . , end table

The width of the rectangle is 13 ft.

Exercises

Solve each equation for x.

30. a x + b x = − c eh x plus b x equals negative c
31. x + r t + 1 = 0 fraction x plus r , over t end fraction . plus 1 equals 0
32. m − 3 x = 2 x + p m minus 3 x equals 2 x plus p
33. x p + x q = s x over p , plus , x over q , equals s

Solve each problem. Round to the nearest tenth, if necessary. Use 3.14 for π.

34. What is the width of a rectangle with length 5.5 cm and area 220  cm 2 ? 220 , cm squared , question mark
35. What is the radius of a circle with circumference 94.2 mm?
36. A triangle has height 15 in. and area 120  in. 2 . 120 . in. squared , .  What is the length of its base?

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