Chapter 7 Lesson 2Chapter 7 Lesson 2Solving Equations with Solving Equations with

Grouping SymbolsGrouping Symbolspgs. 334-338pgs. 334-338

What you will learn:What you will learn:

Solve equations that involve grouping Solve equations that involve grouping symbolssymbols

Identify equations that have no solution Identify equations that have no solution or an infinite number of solutionsor an infinite number of solutions

Vocabulary

• Null/empty set (336): equations that have no solution. No value

of the variable results in a true sentence. Represented by or { }

• Identity (336): an equation that is true for every value of the variable

Josh starts walking at a rate of 2 mph. One hour later, his sister Maria starts on

the same path on her bike, riding at 10mph

Rate

(mph)

Time

(hours)

Distance

(miles)

Josh 2 t 2t

Maria 10 t-1 10(t-1)

What does t represent?

Why is Maria’s time shown as t-1?

Write an equation that represents the time when Maria catches up to Josh.

The time Josh travels

She left 1 hour later than Josh

2t = 10(t-1)

Example 1: Solve Equations with Parentheses

Solve the equation from the previous chart: 2t = 10(t-1)

Write the problem: 2t = 10(t -1)Distributive Property: 2t = 10(t) - 10(1)

Simplify: 2t = 10t -10Subtract 10t from each side: 2t -10t = 10t - 10t -10

Simplify: -8t = -10

Divide each side by -8: -8t = -10 -8 -8

Simplify/Solve: t = 5 or 1 1/4 4

Now check the previous problem.

Josh traveled 2 miles 5 hour or 2.5 miles hour 4Maria traveled 1 hour less than Josh. She

traveled: 10 miles 1 hour hour 4 or 2.5 miles Therefore, Maria caught up to Josh in 1/4

hour or 15 minutes.

Another Example 1: Solve Equations with Parentheses

Solve: 6(n-3) = 4(n + 2.1)

Distributive Property on both sides: 6(n) - 6(3) = 4(n) + 4(2.1)

Simplify: 6n - 18 = 4n + 8.4

Subtract 4n from each side: 6n -4n -18 = 4n -4n + 8.4

Simplify: 2n - 18 = 8.4

Add 18 to both sides: 2n - 18 +18 = 8.4 + 18

Simplify: 2n = 26.4

Divide both sides by 2: 2n = 26.4 2 2

Simplify/Solve: n = 13.2

Check your solution!

Example 2: Use an Equation to Solve a Problem

The perimeter of a rectangle is 20 feet. The width is 4 feet less than the length. Find the dimensions of the rectangle. Then find its area.

Words: The width is 4 feet less than the length. The perimeteris 20 feet.

Symbols: Let A = area Let L-4 = width

2length + 2width = perimeter

Equation: 2length + 2(L-4) = 20 2L + 2L - 8 = 20 4L - 8 = 20 4L = 28 L = 7

• Since we know the length is 7 ft, now we need to find the width.

Formula: 2L + 2W = Perimeter 2(7) + 2W = 20 14 + 2W = 20 2W = 6

W = 3 So the width is 3 feet

Check: 2(7) + 2(3) = 20 14 + 6 = 20

20 = 20

Now find the areaOf the rectangle.A = LWA = 73A = 21 ft2

Example 3: No Solution

Solve: 12 - h = -h + 3

Add an h to both sides: 12 - h + h = -h + h + 3

Simplify: 12 = 3

The sentence 12 = 3 is never true. So the Solution set is

Example 4: All Numbers as solutions

Remember, an equation that is true for every value of the variable is

called an identity.

Solve: 3(2g + 4) = 6(g+2)Distributive Property: 3(2g) + 3(4) = 6(g) + 6(2)

Simplify: 6g + 12 = 6g + 12

Subtract 12 from each side: 6g + 12 - 12 = 6g + 12 - 12

Simplify: 6g = 6g

Mentally divide each side by 6: g = gThe sentence g = g is always true, the solution setis all numbers.

Your Turn!Solve each equation. Check your

solution

A. 3(a-5) = 18

B. 3(s+22) = 4(s+12)

C. 4(f+3) + 5 = 17 + 4f

D. 8y - 3 = 5(y - 1) +3y

a = 11 Check: 3(11-5) = 18 3(6) = 18 18 = 18 S = 18 Check: 3(18+22) = 4(18+12) 3(40) = 4(30)

120 = 120

f = f The solution set is all numbers

The solution set is

One More!

Find the dimension of the rectangle.

P = 460ft

w

w + 30

2(w) + 2(w+30)=4602w + 2w + 60 = 4604w + 60 = 4604w + 60-60 = 460 -604w = 400w =100

w + 30 =Length100 + 30 = Length130 = L

So the the rectangle is 130ft by 100ft

• PRACTICE IS BY THE DOOR ON YOUR WAY OUT!

• QUIZ TOMORROW OVER 7-1 & 7-2