Math  /  Algebra

QuestionMixed Review Solve each equation.
60. 2a+5a4=11-2 a+5 a-4=11
61. 6=3(x+4)6=-3(x+4)
62. 3(c+13)=43\left(c+\frac{1}{3}\right)=4
63. A carpenter is filling in an open entranceway with a door and two side panels of the same width. The entranceway is 3 m wide. The door will be 1.2 m wide. How wide should the carpenter make the panels on either side of the door so that the two panels and the door will fill the entranceway exactly? Get Ready! To prepare for Lesson 2-5, do Exercises 64-66. Evaluate each expression for the given values of the variables.
64. n+2m;m=12,n=2n+2 m ; m=12, n=-2
65. 3b÷c;b=12,c=43 b \div c ; b=12, c=4
66. xy2;x=2.8,y=x y^{2} ; x=2.8, y=

Studdy Solution

STEP 1

What is this asking? We've got a mix of solving equations and evaluating expressions, plus a word problem about a doorway! Watch out! Don't forget to distribute correctly in equations with parentheses and pay close attention to negative signs.
For the word problem, make sure your answer makes real-world sense!

STEP 2

1. Solve equation 60
2. Solve equation 61
3. Solve equation 62
4. Solve the word problem 63
5. Evaluate expression 64
6. Evaluate expression 65
7. Evaluate expression 66

STEP 3

We're asked to solve 2a+5a4=11-2a + 5a - 4 = 11.
First, let's **combine like terms** on the left side: 2a+5a-2a + 5a gives us 3a3a.
So, our equation becomes 3a4=113a - 4 = 11.

STEP 4

Now, we want to **isolate** the term with aa.
We can do this by *adding* **4** to *both* sides of the equation: 3a4+4=11+43a - 4 + 4 = 11 + 4.
This simplifies to 3a=153a = 15.

STEP 5

Finally, to **solve for** aa, we *divide* both sides by **3**: 3a÷3=15÷33a \div 3 = 15 \div 3.
This gives us our **solution**: a=5a = 5!

STEP 6

We have 6=3(x+4)6 = -3(x + 4).
Let's **distribute** the 3-3 on the right side: 6=3x+(3)46 = -3 \cdot x + (-3) \cdot 4, which simplifies to 6=3x126 = -3x - 12.

STEP 7

To **isolate** the xx term, we *add* **12** to both sides: 6+12=3x12+126 + 12 = -3x - 12 + 12.
This becomes 18=3x18 = -3x.

STEP 8

Now, *divide* both sides by 3-3 to **solve for** xx: 18÷(3)=3x÷(3)18 \div (-3) = -3x \div (-3).
So, x=6x = -6.

STEP 9

The equation is 3(c+13)=43(c + \frac{1}{3}) = 4. **Distribute** the 33: 3c+313=43 \cdot c + 3 \cdot \frac{1}{3} = 4, which simplifies to 3c+1=43c + 1 = 4.

STEP 10

*Subtract* **1** from both sides: 3c+11=413c + 1 - 1 = 4 - 1, giving us 3c=33c = 3.

STEP 11

*Divide* both sides by **3**: 3c÷3=3÷33c \div 3 = 3 \div 3.
Therefore, c=1c = 1.

STEP 12

The entranceway is **3 m** wide, and the door is **1.2 m** wide.
Let ww be the width of *one* side panel.
The combined width of the door and *two* panels equals the total width of the entranceway.

STEP 13

We can write this as an equation: 1.2+2w=31.2 + 2w = 3.
To **isolate** the term with ww, *subtract* **1.2** from both sides: 1.21.2+2w=31.21.2 - 1.2 + 2w = 3 - 1.2, which simplifies to 2w=1.82w = 1.8.

STEP 14

*Divide* both sides by **2** to find the width of one panel: 2w÷2=1.8÷22w \div 2 = 1.8 \div 2.
So, w=0.9w = 0.9.
Each panel should be **0.9 m** wide.

STEP 15

We have n+2mn + 2m, with m=12m = 12 and n=2n = -2. **Substitute** these values: 2+212-2 + 2 \cdot 12.

STEP 16

Following the order of operations, multiply first: 2+24-2 + 24.
Then, add to get the **final result**: 2222.

STEP 17

The expression is 3b÷c3b \div c, with b=12b = 12 and c=4c = 4. **Substitute**: 312÷43 \cdot 12 \div 4.

STEP 18

Multiply: 36÷436 \div 4.
Then, divide to get 99.

STEP 19

We have xy2xy^2, with x=2.8x = 2.8 and y=1.5y = 1.5. **Substitute**: 2.8(1.5)22.8 \cdot (1.5)^2.

STEP 20

First, calculate 1.52=1.51.5=2.251.5^2 = 1.5 \cdot 1.5 = 2.25.
Then, 2.82.25=6.32.8 \cdot 2.25 = 6.3.

STEP 21

60. a=5a = 5
61. x=6x = -6
62. c=1c = 1
63. The panels should be 0.9 m wide.
64. 22
65. 9
66. 6.3

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