4.4 Equations of the Form ax + b = cx + d
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- Sharyl Warner
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1 4.4 Equations of the Form ax + b = cx + d We continue our study of equations in which the variable appears on both sides of the equation. Suppose we are given the equation: 3x + 4 = 5x! 6 Our first step is to convert the equation to one of the form ax + b = c, from the last section. Note that the Addition property of equations allows us to add the same quantity to each side of the equation. This quantity can also be an expression (rather than a number). To eliminate the 5x term on the right-hand side of the equation, we will add its opposite 5x to each side of the equation: 3x + (!5x) + 4 = 5x + (!5x)! 6!x + 4 =!6 We can now solve the equation as we did in the previous section:!x + 4 =!6!x (!4) =!6 + (!4)!x =!0! (!x) =! (!0) x = 5 To check our solution, we substitute x = 5 into both sides of the equation: = 5 5! = 5! 6 9 = 9 3
2 Note that we could have also solved the equation by adding 3x (the opposite of 3x) to each side of the equation: 3x + (!3x) + 4 = 5x + (!3x)! 6 4 = x! = x! = x 0 = x 5 = x Note that the solution is the same (although the variable appears on the right-hand side). Generally we will add the variable to the left-hand side, primarily for consistency with the last section. Example Solve the following equations. Include a check of your solution. a. 5x + 3 = x! 9 b. 4y! 5 =!y! 7 c.!3x + = x! 6 d. a! 3 = 5a! 7 Solution a. First add x to each side of the equation, then solve the resulting equation: 5x + 3 = x! 9 ( ) + 3 = x + (!x)! 9 5x +!x 3x + 3 =!9 3x (!3) =!9 + (!3) 3x =! 3 3x = 3 (!) x =!4 5!4 ( ) + 3 =!4 ( )! 9!0 + 3 =!8! 9!7 =!7 33
3 b. First add y to each side of the equation, then solve the resulting equation: 4y! 5 =!y! 7 4y + y! 5 =!y + y! 7 6y! 5 =!7 6y! =! y =! 6 6y = 6 (!) y =! 4! ( )! 5 =!! ( )! 7!8! 5 = 4! 7!3 =!3 c. First add x to each side of the equation, then solve the resulting equation:!3x + = x! 6!3x +!x ( ) + = x + (!x)! 6!5x + =!6!5x + +! ( ) =!6 + (!)!5x =!8! 5 (!5x) =! 5 (!8) x = 8 5! = 8 5! 6! = 6 5! 6! = 6 5! 30 5! 4 5 =!
4 d. First add 5a to each side of the equation, then solve the resulting equation: a! 3 = 5a! 7 ( )! 3 = 5a + (!5a)! 7 a +!5a!3a! 3 =!7!3a! =!7 + 3!3a =!4! 3 (!3a) =! 3 (!4) a = ! 3 = 5 4 3! 7 8 3! 3 = 0 3! 7 8 3! 9 3 = 0 3! 3! 3 =! 3 As in the previous section, equations which contain fractions are best solved by multiplying by the LCM of all fractions within the equation. The next example illustrates this technique further. Example Solve each equation by first eliminating fractions. Include a check of your solution. a. x = x! 3 b. 3 y! = y! 3 c. 6 s! 3 4 = 4 s! 3 d..4t!.3 = 4.4t!
5 Solution a. The LCM of, 4, and 3 is. Multiplying each side of the equation by and solving the resulting equation: x = x! 3 x + 3 % 4 x! % 3 x = x! 3 6x + 9 = x! 8 ( ) + 9 = x + (!x)! 8 6x +!x!6x + 9 =!8!6x + 9! 9 =!8! 9!6x =!7! 6 (!6x) =! 6 (!7) x = = 7 6! = 7 6! = 7 6! = =
6 b. The LCM of 3 and is 6. Multiplying each side of the equation by 6 and solving the resulting equation: 3 y! = y! y! % = 6 y! % y! 6 = 6y! 6 3 y! 3 = 6y! ( )! 3 = 6y + (!6y)! y +!6y!4y! 3 =!!4y! =! + 3!4y =! 4 (!4y) =! 4 y =! 4 3! % 4! =! 4! 3!! =! 4! 3!! 6 =! 3! 4! 7 =! 7 37
7 c. The LCM of 6, 4, and 3 is. Multiplying each side of the equation by and solving the resulting equation: 6 s! 3 4 = 4 s! 3 6 s! 3 % 4 = 4 s! % 3 6 s! 3 4 = 4 s! 3 s +!3s s! 9 = 3s! 8 ( )! 9 = 3s + (!3s)! 8!s! 9 =!8!s! =!8 + 9!!s!s = ( ) =! s =! 6! ( )! 3 4 = 4! ( )! 3! 6! 3 4 =! 4! 3!! 9 =! 3! 8! =! 38
8 d. This equation may not appear to contain fractions, but remember that the decimals in this equation each represent tenths. Multiplying each side of the equation by 0 and solving the resulting equation:.4t!.3 = 4.4t! 5.6 ( ) = 0( 4.4t! 5.6) 0.4t!.3 0.4t! 0.3 = 0 4.4t! t +!44t 4t! 3 = 44t! 56 ( )! 3 = 44t + (!44t)! 56!0t! 3 =!56!0t! =!56 + 3!0t =!43! 0 (!0t) =! 0 (!43) t = 43 0 t =.5 Note that we converted the fractional answer to a decimal, since the original equation involved decimals..4.5!.3 = 4.4.5! !.3 = 9.46! = 3.86 We now are armed with the techniques for solving all equations of the form ax + b = cx + d. Recall in Section 4. we simplified expressions by combining like terms. We now include some of those expressions into the form of equations. Suppose we are given the equation: 3( x! 4) =!( x + 4) Begin by simplifying each side of the equation (using the distributive property): ( ) =!( x + 4) 3 x! 4 3 x! 3 4 =!x + (!) 4 6x! =!x! 8 39
9 Now solve the equation as in the previous example: 6x! =!x! 8 6x + x! =!x + x! 8 8x! =!8 8x! + =!8 + 8x = 4 8 8x = 8 4 x = 3! 4 % =! + 4 % ( ) =! + 8 3! 4 3!3 ( ) =! 9!9 =!9 The next example will provide additional practice in solving these equations. % % Example 3 Solve the following equations. Include a check of your solution. ( ) = 3( x + ) ( ) = 3( 3y + ) ( ) = 3( b + ) ( 5w! 4) = ( 3 w + ) a. 5 x! b. 4 y! c.! 3b! d. 30
10 Solution a. Apply the distributive property then solve the resulting equation: 5 x! 5x +!6x ( ) = 3( x + ) 5x! 0 = 6x + 3 ( )! 0 = 6x + (!6x) + 3!x! 0 = 3!x! = 3 + 0!!x 5!3!!x = 3 ( ) =! 3 x =!3 ( ) = 3( (!3) + ) 5 (!5) = 3 (!6 + )!75 = 3 (!5)!75 =!75 b. Apply the distributive property then solve the resulting equation: 4 y! ( ) = 3( 3y + ) 8y! 4 = 9y + 6 ( )! 4 = 9y + (!9y) + 6 8y +!9y!y! 4 = 6!y! = 6 + 4!!y!y = 0 ( ) =! 0 y =!0 4 (!0)! ( ) = 3( 3 (!0) + ) 4 (!0! ) = 3 (!30 + ) 4 (!) = 3 (!8)!84 =!84 3
11 c. Apply the distributive property then solve the resulting equation:! 3b!!6b +!6b ( ) = 3( b + )!6b + 4 = 6b + 3 ( ) + 4 = 6b + (!6b) + 3!b + 4 = 3!b (!4) = 3 + (!4)!b =!! (!b) =! (!) b =! 3! % = 3 + %! 4! % = %! 4! 8 % 4 = % 6!! 7 % 4 = 3 7 % 6 7 = 7 3
12 d. Apply the distributive property, clear the equation of fractions by multiplying by the LCM, then solve the resulting equation: 5w! 4 ( ) = 3 w + ( ) 5 w! = 3 w w! % = 6 3 w + % w! 6 = 6 3 w w! = 4w + 5w +!4w ( )! = 4w + (!4w) + w! = w! + = + w = 4 w = 4 w = 4 5 4! 4 % = % 70! 4 % = % 70! 44 % = % 6% = 39% 3 3 = 3 33
13 Not all equations we encounter have unique solutions. Consider the equation: 5x! 3 = 5x + To isolate the variable, we add!5x to each side of the equation: 5x +!5x 5x! 3 = 5x + ( )! 3 = 5x + (!5x) +!3 = But note this equation is false. Recall that solutions are values of the variables which result in a true statement, and thus there is no solution to this equation. Suppose the equation to solve is: 4x + 6 = ( x + 3) Using the distributive property and solving the equation: 4x + 6 = ( x + 3) 4x + 6 = 4x + 6 ( ) + 6 = 4x + (!4x) + 6 4x +!4x 6 = 6 Again note that the variable dropped out of the equation, but this time the equation is true. Thus any value for the variable results in a true statement, so we say any number is a solution to this equation. Example 4 Solve each equation. State that there is no solution or any number solution. a. x! 3 = x! 3 b. 3 4r! 5 ( ) = ( 6r! 7) c. y! % 3 = 6 y! % 3 d. 5 3 v! 3 % 5 = 0 v! % 34
14 Solution a. Solving the equation: x! 3 = x! 3 ( )! 3 = x + (!x)! 3 x +!x!3 =!3 Since this statement is true for any value of x, any number is a solution. b. Using the distributive property and solving the equation: 3 4r! 5 r +!r ( ) = ( 6r! 7) r! 5 = r! 4 ( )! 5 = r + (!r)! 4!5 =!4 Since this statement is false, there is no solution to this equation. c. Using the distributive property and solving the equation: y! % 3 = 6 y! % 3 y! 3 = 6y! 6 3 6y! 4 = 6y! 4 ( )! 4 = 6y + (!6y)! 4 6y +!6y!4 =!4 Since this statement is true for any value of y, any number is a solution. d. Using the distributive property and solving the equation: 5 3 v! 3 % 5 = 0 v! % 5 3 v! = 0 v! 0 5v! 9 = 5v! 0 ( )! 9 = 5v + (!5v)! 0 5v +!5v!9 =!0 Since this statement is false, there is no solution to this equation. 35
15 Example 5 Solve the following equations. State if there is no solution or any number solution. Include a check of your solution. ( )! 4( x! ) =!3 ( )! ( 3y + ) = ( y! ) ( )! 4( s! 3) = 5( s + )! 4( 3s + ) ( 3 t + )! ( t! ) =! a. 5 x + b. 4y! 3 c.!3 s + 4 d. Solution a. Applying the distributive property, then simplifying and solving the equation: 5!6 + 5( x + )! 4( x! ) =!3 5x + 5! 4x + 8 =!3 x + 3 =!3 x (!3) =!3 + (!3) x =!6 ( )! 4 (!6! ) =!3 5 (!5)! 4 (!8) =!3! =!3!3 =!3 b. Applying the distributive property, then simplifying and solving the equation: 4y! 3 ( )! ( 3y + ) = ( y! ) 8y! 6! 6y! = y! 4 y +!y y! 8 = y! 4 ( )! 8 = y + (!y)! 4!8 =!4 Since this statement is false, there is no solution to the equation. 36
16 c. Applying the distributive property, then simplifying and solving the equation:!3 s + 4 ( )! 4 s! 3 ( ) = 5 s + ( )! 4 3s + ( )!3s!! 4s + = 0s + 5! s! 4!7s =!s +!7s + s =!s + s +!5s =! 5 (!5s) =! 5 s =! 5!3! %! 4! 5! 3 % = 5! % % 5 +! 4 3! % % 5 +!3! %! 4! 5! 3 % = 5! 5 + %! 4! %!3! % 5! 4! 5! 5 % 5 = 5! % 5! 4! % 5!3 9 % 5! 4! 6 % 5 = 5 3 % 5! 4 % 5! = 5 5! =
17 d. Applying the distributive property, then simplifying and solving the equation: 3 t + ( )! t! ( ) =! 3 t + 3! t + =! 3 t + 3! t + % =! % 3 t + 3! t + =!6 8t + 8! 6t + =!6 t + 0 =!6 t =!6 t = (!6) t =!3 ( 3!3 + )! (!3! ) =! ( 3!)! (!5) =!!8 + 5 =!! =!! =! In the next section we will consider applications of the algebraic equations we studied in these two sections. Be sure that you understand how to solve these equations through the exercises in this section. 38
18 Exercise Set 4.4 Solve the following equations. State if there is no solution or any number solution. Include a check of your solution.. 4x + 5 = 3x x + 6 = 7x x + 7 = 6x x + 5 = 8x y + 8 = y! y + 9 = 3y! y! 6 = 3y! y! 7 = 7y! 9. 4a! = 7a a! 9 = 8a + 6.!a + 3 = a!.!5a + =!3a! 5 3.!6b + =!3b! 7 4.!7b + =!3b! b! 3 =!3 + 8b 6. b! 5 = 4 + b 7.!3s! 6 = 6! 3s 8.!4s! 9 =!9! 4s 9.!6s + 5 = 8! 3s 0.!7s + = 5! 5s. 4t + 5 = 6t! 7. 7t! 8 = t + 3.!3t + 7 = 5t! 3 4.!t + 6 = 4t! Solve each equation by first eliminating fractions and/or decimals. Include a check of your solution x + 4 = x! 6. 3 x! = x! y + 3 = y! y + = y! s + = 3 5 s! s = 5 s! 3. t + 3 = 6 t! t = t! a! 3 = 3 4 a a = a! b!.5 = 3.3b! b!.9 =.5b! 37.!.r! 3.4 =!.9r!.5 38.!.7r! 4.7 =!.r! t = t =
19 Solve the following equations. State if there is no solution or any number solution. Include a check of your solution x + 3 ( ) = 3( x! 3) 4. 7( x! ) = 3( x! 5) ( ) = ( 4y! 3) 44. ( y! ) = 4( 3y! 5) ( ) = ( 3 a! 5) ( ) = ( 3 a! 3) y! 45. a! a! 5 ( 8 3b! ) = 3 ( 4 b! 3) ( 6 b! 3) = ( 3 b! 5) 49.! 5 ( 6 s! ) =! ( 3 4s! ) 50.! ( 3s! ) =! 3 ( 4 6s! 5) 5. 6 t! 5 53.!4 3u! v! w! x r! y! 65.!3 a! 67.!4 5b! ( ) = 3( 4t! 0) 5. 5( 4t! 3) = ( 0t! 7) ( ) =!3( 4u! 6) 54.!3( u! 4) =!( 3u! 6) ( ) = 0.4( v! 3) 56.!0.4( v + 7) = 0.( v! 5) ( ) = 3( 3.w!.) 58. 5(.5w!.) = 4(.9w!.5) ( )! 6( x + 3) =!9 60. ( x! 3)! ( x! 4) =!8 ( )! ( r + ) =! 6. 5( r! 4)! 8( r! 3) =!8 ( )! 4( 3y! ) =! ( 4y! 3)! ( 0y! 6) =!3 ( ) + ( 4a! 3) =!( 3a! ) 66.!4( a! 3) + 3( 3a! ) =!( a! 3) ( ) + 3( b! ) =!5( 3b! 4) 68.!3( b! 5) + ( 4b! 3) =!( b! 5) 3 v + ) 6 v! ) 70.! 3 4 v! ) 3 v + 3) 3 7.! 5 ( 6 s! )! ( s + ) =! !( g + 3)! 5( g! ) = 4 3g! 74.!3( p + 4)! 4( p! ) = 3 4 p! 5 75.!5( x + 3)! 4( 3x! ) = 5 x! 76.!3( 4y! )! 3( y! 3) =!4 y! 3 ( )! 3( g! ) ( )! 5( 3p! 4) ( )! 4( 3x! ) ( )! 3( y! ) 7.! ( 6 s! )! ( 3 s + ) =! 330
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