solve multi - step linear equations
1. elsie is planting a rectangular section of grass.
if the perimeter of the rectangle is 96 feet, what are the length and width of the rectangular section?
2. math on the spot solve each equation.
a. 3n + 1 = 19
b. 21 = - 2p - 5
3. reason what step would you perform first to solve the following equation? explain your reasoning.
$\frac{1}{4}(12 - 8x)=\frac{2}{3}(6x)$
4. heather has a family phone plan. the monthly payment for the family plan includes a $70 charge for unlimited talk and text, a $20 line fee per phone, and a $22.91 equipment fee for each phone. their total monthly bill is $241.64. write and solve an equation to find how many phones are on the plan.
solve each equation. check your solution.
5. a + 3(a - 1)=3(2 + 1)
6. 5y - 3(2 - y)=10
7. 1.2x - 2 = 7 + 0.9x
8. - k + 4(k + 1)=2k
9. 4($\frac{x}{6}$ + 5)=2x + 10
10. 3w+$\frac{w}{2}$ + 1 = 10 - w

Question

Accepted Answer

### Explanation:
#### Step1: Recall perimeter formula
The perimeter formula for a rectangle is $P = 2(l + w)$. Here, $l=6x - 15$, $w = x$ and $P=96$. So, $96=2((6x - 15)+x)$.
#### Step2: Simplify the equation
First, simplify the expression inside the parentheses: $96 = 2(7x - 15)$. Then distribute the 2: $96=14x-30$.
#### Step3: Solve for x
Add 30 to both sides of the equation: $96 + 30=14x-30 + 30$, which gives $126 = 14x$. Then divide both sides by 14: $x=\frac{126}{14}=9$.
#### Step4: Find length and width
Width $w=x = 9$ feet. Length $l=6x - 15=6	imes9 - 15=54 - 15 = 39$ feet.

### Answer:
Length is 39 feet and width is 9 feet.

### Explanation for 2A:
#### Step1: Isolate the term with n
Subtract 1 from both sides of the equation $3n + 1=19$. We get $3n=19 - 1=18$.
#### Step2: Solve for n
Divide both sides by 3: $n=\frac{18}{3}=6$.

### Answer for 2A:
$n = 6$

### Explanation for 2B:
#### Step1: Isolate the term with p
Add 5 to both sides of the equation $21=-2p - 5$. We get $21 + 5=-2p$, so $26=-2p$.
#### Step2: Solve for p
Divide both sides by - 2: $p=\frac{26}{-2}=-13$.

### Answer for 2B:
$p=-13$

### Explanation for 3:
#### Step1: Simplify both sides
First, simplify $\frac{1}{4}(12 - 8x)$ to $3 - 2x$ and $\frac{2}{3}(6x)$ to $4x$. The equation becomes $3-2x = 4x$. The first - step is to simplify the expressions on both sides of the equation using the distributive property because it makes the equation easier to solve by getting rid of the fractions and parentheses.

### Answer for 3:
Use the distributive property to simplify both sides of the equation.

### Explanation for 4:
#### Step1: Set up the equation
Let the number of phones be $x$. The monthly payment equation is $70+20x + 22.91x=241.64$. Combine like - terms: $70+(20 + 22.91)x=241.64$, so $70+42.91x=241.64$.
#### Step2: Solve for x
Subtract 70 from both sides: $42.91x=241.64 - 70=171.64$. Then divide both sides by 42.91: $x=\frac{171.64}{42.91}=4$.

### Answer for 4:
There are 4 phones on the plan.

### Explanation for 5:
#### Step1: Expand the equation
Expand $a + 3(a - 1)=3(2 + 1)$. First, expand $3(a - 1)$ to $3a-3$ and $3(2 + 1)$ to 9. The equation becomes $a+3a - 3=9$.
#### Step2: Combine like - terms
Combine the a terms: $4a-3 = 9$.
#### Step3: Solve for a
Add 3 to both sides: $4a=9 + 3=12$. Then divide both sides by 4: $a = 3$.

### Answer for 5:
$a = 3$

### Explanation for 6:
#### Step1: Expand the equation
Expand $5y-3(2 - y)=10$. Expand $-3(2 - y)$ to $-6 + 3y$. The equation becomes $5y-6 + 3y=10$.
#### Step2: Combine like - terms
Combine the y terms: $8y-6 = 10$.
#### Step3: Solve for y
Add 6 to both sides: $8y=10 + 6=16$. Then divide both sides by 8: $y = 2$.

### Answer for 6:
$y = 2$

### Explanation for 7:
#### Step1: Isolate the terms with x
Subtract $0.9x$ from both sides of the equation $1.2x-2=7 + 0.9x$. We get $1.2x-0.9x-2=7$, so $0.3x-2=7$.
#### Step2: Solve for x
Add 2 to both sides: $0.3x=7 + 2=9$. Then divide both sides by 0.3: $x=\frac{9}{0.3}=30$.

### Answer for 7:
$x = 30$

### Explanation for 8:
#### Step1: Expand the equation
Expand $-k + 4(k + 1)=2k$. Expand $4(k + 1)$ to $4k+4$. The equation becomes $-k+4k + 4=2k$.
#### Step2: Combine like - terms
Combine the k terms: $3k+4=2k$.
#### Step3: Solve for k
Subtract $3k$ from both sides: $4=2k-3k=-k$. So, $k=-4$.

### Answer for 8:
$k=-4$

### Explanation for 9:
#### Step1: Expand the equation
Expand $4(\frac{x}{6}+5)=2x + 10$. Expand $4(\frac{x}{6}+5)$ to $\frac{4x}{6}+20=\frac{2x}{3}+20$. The equation becomes $\frac{2x}{3}+20=2x + 10$.
#### Step2: Get rid of the fraction
Multiply through by 3 to get $2x+60 = 6x+30$.
#### Step3: Solve for x
Subtract $2x$ from both sides: $60=6x - 2x+30$, so $60 = 4x+30$. Subtract 30 from both sides: $30 = 4x$. Divide both sides by 4: $x=\frac{30}{4}=7.5$.

### Answer for 9:
$x = 7.5$

### Explanation for 10:
#### Step1: Combine like - terms
Combine the w terms in $3w+\frac{w}{2}+1=10 - w$. First, get a common denominator for the w terms on the left - hand side. $3w+\frac{w}{2}=\frac{6w + w}{2}=\frac{7w}{2}$. The equation becomes $\frac{7w}{2}+1=10 - w$.
#### Step2: Get rid of the fraction
Multiply through by 2: $7w+2 = 20-2w$.
#### Step3: Solve for w
Add $2w$ to both sides: $7w+2w+2=20$, so $9w+2=20$. Subtract 2 from both sides: $9w=18$. Divide both sides by 9: $w = 2$.

### Answer for 10:
$w = 2$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall perimeter formula

Step2: Simplify the equation

Step3: Solve for x

Step4: Find length and width

Step1: Isolate the term with n

Step2: Solve for n

Step1: Isolate the term with p

Step2: Solve for p

Step1: Simplify both sides

Answer: