2-4 solving equations with variables on both sides warm up warm up lesson presentation lesson...
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2-4 Solving Equations with Variables on Both Sides
Warm UpWarm Up
Lesson Presentation
California Standards
PreviewPreview
2-4 Solving Equations with Variables on Both Sides
Warm UpSimplify.
1. 4x – 10x
2. –7(x – 3)
3.
4. 15 – (x – 2)
Solve.
5. 3x + 2 = 8
6.
–6x–7x + 21
17 – x
2x + 3
2
28
2-4 Solving Equations with Variables on Both Sides
4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12.
5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
California Standards
2-4 Solving Equations with Variables on Both Sides
Vocabulary
identity
2-4 Solving Equations with Variables on Both Sides
To solve an equation with variables on both sides, use inverse operations to "collect" variable terms on one side of the equation.
Helpful HintEquations are often easier to solve when the variable has a positive coefficient. Keep this in mind when deciding on which side to "collect" variable terms.
2-4 Solving Equations with Variables on Both Sides
Solve 7n – 2 = 5n + 6.
Additional Example 1: Solving Equations with Variables on Both Sides
To collect the variable terms on one side, subtract 5n from both sides.
7n – 2 = 5n + 6
–5n –5n
2n – 2 = 6
Since n is multiplied by 2, divide both sides by 2 to undo the multiplication.
2n = 8
+ 2 + 2
n = 4
Since 2 is subtracted from 2n, add 2 to both sides.
2-4 Solving Equations with Variables on Both Sides
Solve the equation. Check your answer.
Check It Out! Example 1a
To collect the variable terms on one side, subtract 3b from both sides.
4b + 2 = 3b
–3b –3b
b + 2 = 0
b = –2
– 2 – 2
4b + 2 = 3b
Since 2 is added to b, subtract 2 from both sides.
2-4 Solving Equations with Variables on Both Sides
Check It Out! Example 1a Continued
Solve the equation. Check your answer.
4b + 2 = 3b CheckTo check your answer,
substitute –2 for b. 4(–2) + 2 3(–2)
–8 + 2 –6
–6 –6
2-4 Solving Equations with Variables on Both Sides
0.5 + 0.3y = 0.7y – 0.3
Check It Out! Example 1b
To collect the variable terms on one side, subtract 0.3y from both sides.
0.5 + 0.3y = 0.7y – 0.3
–0.3y –0.3y
0.5 = 0.4y – 0.3
0.8 = 0.4y
+0.3 + 0.3
2 = y
Since 0.3 is subtracted from 0.4y, add 0.3 to both sides to undo the subtraction.
Since y is multiplied by 0.4, divide both sides by 0.4 to undo the multiplication.
Solve the equation. Check your answer.
2-4 Solving Equations with Variables on Both Sides
Check It Out! Example 1b Continued
Solve the equation. Check your answer.
0.5 + 0.3y = 0.7y – 0.3
To check your answer, substitute 2 for y.
Check
0.5 + 0.3(2) 0.7(2) – 0.3
1.1 1.1 0.5 + 0.6 1.4 – 0.3
2-4 Solving Equations with Variables on Both Sides
To solve more complicated equations, you may need to first simplify by using the Distributive Property or combining like terms.
2-4 Solving Equations with Variables on Both Sides
Solve the equation.
Additional Example 2: Simplifying Each Side BeforeSolving Equations
Combine like terms.
Distribute –5 to the expression in parentheses.
4 – 6a + 4a = –1 –5(7 – 2a)
4 – 6a + 4a = –1 –5(7) –5(–2a)
4 – 6a + 4a = –1 – 35 + 10a
4 – 2a = –36 + 10a
+36 +36
40 – 2a = 10a
Since –36 is added to 10a, add 36 to both sides.
4 – 6a + 4a = –1 – 5(7 – 2a)
2-4 Solving Equations with Variables on Both Sides
Solve 4 – 6a + 4a = –1 – 5(7 – 2a).
Additional Example 2 Continued
Since a is multiplied by 12, divide both sides by 12.
40 – 2a = 10a+ 2a +2a
40 = 12a
To collect the variable terms on one side, add 2a to both sides.
2-4 Solving Equations with Variables on Both Sides
Check It Out! Example 2a
Since 1 is subtracted from b, add 1 to both sides.
Distribute to the expression in parentheses.
12
+ 1 + 13 = b – 1
To collect the variable terms on one side, subtract b from both sides.
12
Solve the equation. Check your answer.
4 = b
2-4 Solving Equations with Variables on Both Sides
Check It Out! Example 2a Continued
Solve the equation. Check your answer.
Check
To check your answer, substitute 4 for b.
5 5
2-4 Solving Equations with Variables on Both Sides
3x + 15 – 9 = 2(x + 2)
Check It Out! Example 2b
Combine like terms.
Distribute 2 to the expression in parentheses.
3x + 15 – 9 = 2(x + 2)
3x + 15 – 9 = 2(x) + 2(2)
3x + 15 – 9 = 2x + 4
3x + 6 = 2x + 4
–2x –2x
x + 6 = 4 – 6 – 6
x = –2
To collect the variable terms on one side, subtract 2x from both sides.
Since 6 is added to x, subtract 6 from both sides.
Solve the equation. Check your answer.
2-4 Solving Equations with Variables on Both Sides
Check It Out! Example 2b Continued
Solve the equation. Check your answer.
Check
To check your answer, substitute –2 for x.
3x + 15 – 9 = 2(x + 2)
3(–2) + 15 – 9 2(–2 + 2)
–6 + 15 – 9 2(0)
0 0
2-4 Solving Equations with Variables on Both Sides
An identity is an equation that is always true, no matter what value is substituted for the variable. The solution set of an identity is all real numbers. Some equations are always false. Their solution sets are empty. In other words, their solution sets contain no elements.
2-4 Solving Equations with Variables on Both Sides
Solve the equation.
Additional Example 3A: Infinitely Many Solutions or No Solutions
Identify like terms.
11 – 5x = 11 – 5x Combine like terms on
the left and the right.
The statement 11 – 5x = 11 – 5x is true for all values of x. The equation 10 – 5x + 1 = 7x + 11 – 12x is an identity. All values of x will make the equation true. In other words, all real numbers are solutions.
10 – 5x + 1 = 7x + 11 – 12x
10 – 5x + 1 = 7x + 11 – 12x
2-4 Solving Equations with Variables on Both Sides
Additional Example 3B: Infinitely Many Solutions or No Solutions
Subtract 13x from both sides.
Identify like terms.
13x – 3 = 13x – 4
–3 = –4
–13x –13x
Combine like terms on the left and the right.
The equation 12x – 3 + x = 5x – 4 + 8x is always false. There is no value of x that will make the equation true. There are no solutions.
Solve the equation.
False statement; the solution set is .
12x – 3 + x = 5x – 4 + 8x
12x – 3 + x = 5x – 4 + 8x
2-4 Solving Equations with Variables on Both Sides
The empty set can be written as or {}.
Writing Math
2-4 Solving Equations with Variables on Both Sides
Solve the equation.
Check It Out! Example 3a
Subtract 3y from both sides.
Identify like terms.4y + 7 – y = 10 + 3y
3y + 7 = 3y + 10
7 = 10
–3y –3y False statement; the
solution set is .
4y + 7 – y = 10 + 3y
The equation 4y + 7 – y = 10 + 3y is always false. There is no value of y that will make the equation true. There are no solutions.
2-4 Solving Equations with Variables on Both Sides
Solve the equation.
Check It Out! Example 3b
Identify like terms.2c + 7 + c = –14 + 3c + 21
3c + 7 = 3c + 7 Combine like terms on the left and the right.
2c + 7 + c = –14 + 3c + 21
The statement 3c + 7 = 3c + 7 is true for all values of c. The equation 2c + 7 + c = –14 + 3c + 21 is an identity. All values of c will make the equation true. In other words, all real numbers are solutions.
2-4 Solving Equations with Variables on Both Sides
Jon and Sara are planting tulip bulbs. Jon has planted 60 bulbs and is planting at a rate of 44 bulbs per hour. Sara has planted 96 bulbs and is planting at a rate of 32 bulbs per hour. In how many hours will Jon and Sara have planted the same number of bulbs? How many bulbs will that be?
Additional Example 4: Application
Person Bulbs
Jon 60 bulbs plus 44 bulbs per hour
Sara 96 bulbs plus 32 bulbs per hour
2-4 Solving Equations with Variables on Both Sides
Additional Example 4 Continued
Let h represent hours, and write expressions for the number of bulbs planted.
60 bulbs
plus
44 bulbs each hour
the same
as
96 bulbs
plus
32 bulbs each hour
When is ?
60 + 44h = 96 + 32h
60 + 44h = 96 + 32h To collect the variable terms on one side, subtract 32h from both sides.
60 + 12h = 96
–32h –32h
2-4 Solving Equations with Variables on Both Sides
Since 60 is added to 12h, subtract 60 from both sides.
60 + 12h = 96–60 – 60
12h = 36Since h is multiplied by 12,
divide both sides by 12 to undo the multiplication.
h = 3
Additional Example 4 Continued
2-4 Solving Equations with Variables on Both Sides
After 3 hours, Jon and Sara will have planted the same number of bulbs. To find how many bulbs they will have planted in 3 hours, evaluate either expression for h = 3:
60 + 44h = 60 + 44(3) = 60 + 132 = 192
96 + 32h = 96 + 32(3) = 96 + 96 = 192
After 3 hours, Jon and Sara will each have planted 192 bulbs.
Additional Example 4 Continued
2-4 Solving Equations with Variables on Both Sides
Four times Greg's age, decreased by 3 is equal to 3 times Greg's age, increased by 7. How old is Greg?
Check It Out! Example 4
Let g represent Greg's age, and write expressions for his age.
Four times Greg's
age
decreased by
3is
equal to
three times Greg's
age
increased by
7 .
4g – 3 = 3g + 7
2-4 Solving Equations with Variables on Both Sides
Check It Out! Example 4 Continued
4g – 3 = 3g + 7 To collect the variable terms on one side, subtract 3g from both sides.
g – 3 = 7
–3g –3g
Since 3 is subtracted from g, add 3 to both sides.
+ 3 + 3
g = 10
Greg is 10 years old.
2-4 Solving Equations with Variables on Both Sides
Lesson QuizSolve each equation.
1. 7x + 2 = 5x + 8 2. 4(2x – 5) = 5x + 4
3. 6 – 7(a + 1) = –3(2 – a)
4. 4(3x + 1) – 7x = 6 + 5x – 2
5.
6. A painting company charges $250 base plus $16 per hour. Another painting company charges $210 base plus $18 per hour. How long is a job for which the two companies costs are the same?
3 8
all real numbers
120 hours