example 1
solve each equation. check your solution.
1. 6x - 5 = 7 - 9x
2. -1.6r + 5 = -7.8
3. $\frac{3}{4}-\frac{1}{2}n=\frac{5}{8}$
4. $\frac{5}{6}c+\frac{3}{4}=\frac{11}{12}$
5. 2.2n + 0.8n + 5 = 4n
6. 6y - 5 = -3(2y + 1)
7. -6(2x + 4) + $\frac{1}{2}$(8 + 3x) = -20
8. 7(-1 + 4x) - 12x = 5
9. -4(10 + 3x) - (x + 8) = -9
example 2
solve each problem.
10. reasoning the length of a rectangle is twice the width. find the width if the perimeter is 60 centimeters. define a variable, write an equation, and solve the problem.
11. golf sergio and three friends went golfing. two of the friends rented clubs for $6 each. the total cost of the rented clubs and the green fees was $76. what was the cost of the green fees for each person? define a variable, write an equation, and solve the problem.
example 3
solve each equation or formula for the specified variable.
12. banking the formula for simple interest i is i = prt, where p is the principal, r is the interest rate, and t is time. solve for p.
13. mean the mean a of two numbers, x and y, is given by a = $\frac{x + y}{2}$. solve for y.
14. slope the slope m between two points $(x_1,y_1)$ and $(x_2,y_2)$ is m = $\frac{y_2 - y_1}{x_2 - x_1}$. solve for $y_2$.
15. cylinders the surface area of a cylinder a is given by a = 2$pi$r$^2$ + 2$pi$rh, where r is radius and h is height. solve for h.

Question

Accepted Answer

### 1. Solve the equation $6x - 5=7 - 9x$
# Explanation:
## Step1: Add $9x$ to both sides
Adding $9x$ to both sides of the equation $6x - 5=7 - 9x$ gives $6x+9x - 5=7-9x + 9x$, which simplifies to $15x-5 = 7$.
## Step2: Add 5 to both sides
Adding 5 to both sides of $15x - 5=7$ gives $15x-5 + 5=7 + 5$, so $15x=12$.
## Step3: Divide both sides by 15
Dividing both sides of $15x = 12$ by 15 gives $x=\frac{12}{15}=\frac{4}{5}$.
# Answer:
$x=\frac{4}{5}$

### 2. Solve the equation $-1.6r+5=-7.8$
# Explanation:
## Step1: Subtract 5 from both sides
Subtracting 5 from both sides of $-1.6r + 5=-7.8$ gives $-1.6r+5 - 5=-7.8 - 5$, so $-1.6r=-12.8$.
## Step2: Divide both sides by $-1.6$
Dividing both sides of $-1.6r=-12.8$ by $-1.6$ gives $r=\frac{-12.8}{-1.6}=8$.
# Answer:
$r = 8$

### 3. Solve the equation $\frac{3}{4}-\frac{1}{2}n=\frac{5}{8}$
# Explanation:
## Step1: Subtract $\frac{3}{4}$ from both sides
$\frac{3}{4}-\frac{1}{2}n-\frac{3}{4}=\frac{5}{8}-\frac{3}{4}$. Since $\frac{3}{4}=\frac{6}{8}$, we have $-\frac{1}{2}n=\frac{5 - 6}{8}=-\frac{1}{8}$.
## Step2: Multiply both sides by - 2
Multiplying both sides of $-\frac{1}{2}n=-\frac{1}{8}$ by - 2 gives $n=\frac{1}{4}$.
# Answer:
$n=\frac{1}{4}$

### 4. Solve the equation $\frac{5}{6}c+\frac{3}{4}=\frac{11}{12}$
# Explanation:
## Step1: Subtract $\frac{3}{4}$ from both sides
$\frac{5}{6}c+\frac{3}{4}-\frac{3}{4}=\frac{11}{12}-\frac{3}{4}$. Since $\frac{3}{4}=\frac{9}{12}$, we get $\frac{5}{6}c=\frac{11 - 9}{12}=\frac{2}{12}=\frac{1}{6}$.
## Step2: Multiply both sides by $\frac{6}{5}$
Multiplying both sides of $\frac{5}{6}c=\frac{1}{6}$ by $\frac{6}{5}$ gives $c=\frac{1}{6}\times\frac{6}{5}=\frac{1}{5}$.
# Answer:
$c=\frac{1}{5}$

### 5. Solve the equation $2.2n+0.8n + 5=4n$
# Explanation:
## Step1: Combine like - terms on the left side
$(2.2n+0.8n)+5 = 4n$, so $3n+5 = 4n$.
## Step2: Subtract $3n$ from both sides
Subtracting $3n$ from both sides of $3n + 5=4n$ gives $3n-3n + 5=4n-3n$, so $n = 5$.
# Answer:
$n = 5$

### 6. Solve the equation $6y-5=-3(2y + 1)$
# Explanation:
## Step1: Distribute on the right side
$6y-5=-6y-3$.
## Step2: Add $6y$ to both sides
$6y+6y-5=-6y+6y - 3$, so $12y-5=-3$.
## Step3: Add 5 to both sides
$12y-5 + 5=-3 + 5$, so $12y=2$.
## Step4: Divide both sides by 12
$y=\frac{2}{12}=\frac{1}{6}$.
# Answer:
$y=\frac{1}{6}$

### 7. Solve the equation $-6(2x + 4)+\frac{1}{2}(8 + 3x)=-20$
# Explanation:
## Step1: Distribute
$-12x-24 + 4+\frac{3}{2}x=-20$.
## Step2: Combine like - terms
$(-12x+\frac{3}{2}x)+(-24 + 4)=-20$. $-12x+\frac{3}{2}x=\frac{-24x + 3x}{2}=-\frac{21}{2}x$ and $-24 + 4=-20$, so $-\frac{21}{2}x-20=-20$.
## Step3: Add 20 to both sides
$-\frac{21}{2}x-20 + 20=-20 + 20$, so $-\frac{21}{2}x=0$.
## Step4: Multiply both sides by $-\frac{2}{21}$
$x = 0$.
# Answer:
$x = 0$

### 8. Solve the equation $7(-1 + 4x)-12x=5$
# Explanation:
## Step1: Distribute
$-7+28x-12x=5$.
## Step2: Combine like - terms
$(28x-12x)-7=5$, so $16x-7=5$.
## Step3: Add 7 to both sides
$16x-7 + 7=5 + 7$, so $16x=12$.
## Step4: Divide both sides by 16
$x=\frac{12}{16}=\frac{3}{4}$.
# Answer:
$x=\frac{3}{4}$

### 9. Solve the equation $-4(10 + 3x)-(x + 8)=-9$
# Explanation:
## Step1: Distribute
$-40-12x-x - 8=-9$.
## Step2: Combine like - terms
$(-12x-x)+(-40 - 8)=-9$, so $-13x-48=-9$.
## Step3: Add 48 to both sides
$-13x-48 + 48=-9 + 48$, so $-13x=39$.
## Step4: Divide both sides by $-13$
$x=-3$.
# Answer:
$x=-3$

### 10. Solve the rectangle problem
Let the width of the rectangle be $w$ centimeters. The length $l = 2w$ centimeters. The perimeter formula of a rectangle is $P = 2(l + w)$. Given $P = 60$ centimeters, we substitute $l$ and $P$ into the formula:
# Explanation:
## Step1: Substitute into the perimeter formula
$60=2(2w+w)$.
## Step2: Simplify the right - hand side
$60=2(3w)=6w$.
## Step3: Divide both sides by 6
$w = 10$ centimeters.
# Answer:
The width of the rectangle is 10 centimeters.

### 11. Solve the golf problem
Let $g$ be the cost of the green fees for each person. There are 4 people in total. Two friends rented clubs for $\$6$ each, so the cost of club - renting is $2\times6=\$12$. The total cost of club - renting and green fees for 4 people is $12 + 4g$. Given that the total cost is $\$76$.
# Explanation:
## Step1: Write the equation
$12+4g=76$.
## Step2: Subtract 12 from both sides
$12+4g-12=76 - 12$, so $4g=64$.
## Step3: Divide both sides by 4
$g = 16$ dollars.
# Answer:
The cost of the green fees for each person is $\$16$.

### 12. Solve the simple - interest formula for $P$
Given $I = Prt$, to solve for $P$:
# Explanation:
## Step1: Divide both sides by $rt$ (assuming $rt
eq0$)
$P=\frac{I}{rt}$.
# Answer:
$P=\frac{I}{rt}$

### 13. Solve the mean formula for $y$
Given $A=\frac{x + y}{2}$, to solve for $y$:
# Explanation:
## Step1: Multiply both sides by 2
$2A=x + y$.
## Step2: Subtract $x$ from both sides
$y=2A - x$.
# Answer:
$y=2A - x$

### 14. Solve the slope formula for $y_2$
Given $m=\frac{y_2 - y_1}{x_2 - x_1}$, to solve for $y_2$:
# Explanation:
## Step1: Multiply both sides by $(x_2 - x_1)$
$m(x_2 - x_1)=y_2 - y_1$.
## Step2: Add $y_1$ to both sides
$y_2=m(x_2 - x_1)+y_1$.
# Answer:
$y_2=m(x_2 - x_1)+y_1$

### 15. Solve the cylinder - surface - area formula for $h$
Given $A = 2\pi r^{2}+2\pi rh$, to solve for $h$:
# Explanation:
## Step1: Subtract $2\pi r^{2}$ from both sides
$A-2\pi r^{2}=2\pi rh$.
## Step2: Divide both sides by $2\pi r$ (assuming $r
eq0$)
$h=\frac{A - 2\pi r^{2}}{2\pi r}$.
# Answer:
$h=\frac{A - 2\pi r^{2}}{2\pi r}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

1. Solve the equation \(6x - 5=7 - 9x\)

Step1: Add \(9x\) to both sides

Step2: Add 5 to both sides

Step3: Divide both sides by 15

Step1: Subtract 5 from both sides

Step2: Divide both sides by \(-1.6\)

Step1: Subtract \(\frac{3}{4}\) from both sides

Step2: Multiply both sides by - 2

Answer:

2. Solve the equation \(-1.6r+5=-7.8\)