4. $\frac{x}{3} + 6 = 10$\
\
3. linear functions\
1. write the equation of a line that has a slope of 3 and passes through the point (2, 5).\
2. determine whether the following represents a linear function:\
- $y = x^2 - 4x + 2$\
- $y = 3x + 1$\
3. graph the function $y = -2x + 4$. (describe how you would graph it if no graph paper is available.)\
\
4. geometry: area & volume\
1. find the area of a triangle with base 12 cm and height 9 cm.\
2. find the volume of a rectangular prism with dimensions 4 m × 5 m × 3 m.\
3. a circle has a radius of 7 cm. find the circumference and area.\
\
5. word problems\
1. a phone originally costs $950. it goes on sale for 18% off. what is the sale price?\
2. a car travels at an average speed of 62 mph. how far will it travel in 3.5 hours?\
3. a student has scores of 78, 85, 90, and 87 on tests. what score do they need on the fifth test to have an average of 85?\
\
additional reading passages\
passage 1: the power of curiosity\
curiosity has driven human progress for centuries. from the earliest explorers who crossed oceans to modern scientists investigating black holes, the desire to understand the unknown pushes society forward. while curiosity leads to discovery, it also requires discipline. without focus and responsibility, individuals may chase distractions rather than meaningful goals. true curiosity involves asking questions that matter and being willing to put in the effort to find thoughtful answers.

Question

4. $\frac{x}{3} + 6 = 10$\
\
3. linear functions\
1. write the equation of a line that has a slope of 3 and passes through the point (2, 5).\
2. determine whether the following represents a linear function:\
   - $y = x^2 - 4x + 2$\
   - $y = 3x + 1$\
3. graph the function $y = -2x + 4$. (describe how you would graph it if no graph paper is available.)\
\
4. geometry: area & volume\
1. find the area of a triangle with base 12 cm and height 9 cm.\
2. find the volume of a rectangular prism with dimensions 4 m × 5 m × 3 m.\
3. a circle has a radius of 7 cm. find the circumference and area.\
\
5. word problems\
1. a phone originally costs $950. it goes on sale for 18% off. what is the sale price?\
2. a car travels at an average speed of 62 mph. how far will it travel in 3.5 hours?\
3. a student has scores of 78, 85, 90, and 87 on tests. what score do they need on the fifth test to have an average of 85?\
\
additional reading passages\
passage 1: the power of curiosity\
curiosity has driven human progress for centuries. from the earliest explorers who crossed oceans to modern scientists investigating black holes, the desire to understand the unknown pushes society forward. while curiosity leads to discovery, it also requires discipline. without focus and responsibility, individuals may chase distractions rather than meaningful goals. true curiosity involves asking questions that matter and being willing to put in the effort to find thoughtful answers.

Accepted Answer

Since there are multiple sub - questions, we'll solve them one by one using the Step - by - Step Format under the Mathematics discipline (subfields like Algebra for equations, Geometry for area/volume, etc.).

### 1. Solve $\frac{x}{3}+6 = 10$
# Answer:
$x = 12$
# Explanation:
## Step1: Subtract 6 from both sides
To isolate the term with $x$, we subtract 6 from both sides of the equation $\frac{x}{3}+6 = 10$. So we get $\frac{x}{3}+6 - 6=10 - 6$, which simplifies to $\frac{x}{3}=4$.
## Step2: Multiply both sides by 3
To solve for $x$, we multiply both sides of the equation $\frac{x}{3}=4$ by 3. So $x = 4\times3=12$.

### 2. Linear Functions
#### 1. Write the equation of a line with slope $m = 3$ passing through $(2,5)$
# Answer:
$y=3x - 1$
# Explanation:
## Step1: Use the point - slope form
The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(2,5)$ and $m = 3$. Substitute these values into the formula: $y - 5=3(x - 2)$.
## Step2: Simplify the equation
Expand the right - hand side: $y - 5=3x-6$. Then add 5 to both sides: $y=3x - 6 + 5=3x - 1$.

#### 2. Determine if the functions are linear
- For $y=x^{2}-4x + 2$:
# Answer:
Not a linear function
# Explanation:
## Step1: Recall the form of a linear function
A linear function has the form $y = mx + b$, where the highest power of $x$ is 1. In the function $y=x^{2}-4x + 2$, the highest power of $x$ is 2 (from the $x^{2}$ term). So it is a quadratic function, not a linear function.
- For $y = 3x+1$:
# Answer:
Linear function
# Explanation:
## Step1: Recall the form of a linear function
The function $y = 3x + 1$ is in the form $y=mx + b$ with $m = 3$ and $b = 1$, where the highest power of $x$ is 1. So it is a linear function.

#### 3. Graph $y=-2x + 4$
# Answer:
To graph $y=-2x + 4$ (without graph paper):
# Explanation:
## Step1: Find the y - intercept
The y - intercept occurs when $x = 0$. Substitute $x = 0$ into the equation: $y=-2(0)+4 = 4$. So the y - intercept is $(0,4)$.
## Step2: Find the x - intercept
The x - intercept occurs when $y = 0$. Set $y = 0$ in the equation: $0=-2x + 4$. Solve for $x$: $2x=4$, so $x = 2$. The x - intercept is $(2,0)$.
## Step3: Draw the line
Plot the points $(0,4)$ and $(2,0)$ on a coordinate system (mentally or on paper) and draw a straight line through them.

### 3. Geometry: Area & Volume
#### 1. Area of a triangle with base $b = 12$ cm and height $h = 9$ cm
# Answer:
$54\space cm^{2}$
# Explanation:
## Step1: Recall the formula for the area of a triangle
The formula for the area of a triangle is $A=\frac{1}{2}bh$.
## Step2: Substitute the values
Substitute $b = 12$ and $h = 9$ into the formula: $A=\frac{1}{2}\times12\times9 = 6\times9 = 54\space cm^{2}$.

#### 2. Volume of a rectangular prism with dimensions $4\space m\times5\space m\times3\space m$
# Answer:
$60\space m^{3}$
# Explanation:
## Step1: Recall the formula for the volume of a rectangular prism
The formula for the volume of a rectangular prism is $V=l\times w\times h$ (where $l$, $w$, $h$ are length, width, height).
## Step2: Substitute the values
Substitute $l = 4$, $w = 5$, $h = 3$ into the formula: $V=4\times5\times3=60\space m^{3}$.

#### 3. Circumference and area of a circle with radius $r = 7$ cm
- Circumference ($C$):
# Answer:
$C = 14\pi\space cm$ (or approximately $43.98\space cm$)
# Explanation:
## Step1: Recall the formula for circumference
The formula for the circumference of a circle is $C = 2\pi r$.
## Step2: Substitute the value of $r$
Substitute $r = 7$ into the formula: $C=2\pi\times7 = 14\pi\space cm\approx14\times3.14 = 43.98\space cm$.

- Area ($A$):
# Answer:
$A = 49\pi\space cm^{2}$ (or approximately $153.94\space cm^{2}$)
# Explanation:
## Step1: Recall the formula for the area of a circle
The formula for the area of a circle is $A=\pi r^{2}$.
## Step2: Substitute the value of $r$
Substitute $r = 7$ into the formula: $A=\pi\times7^{2}=49\pi\space cm^{2}\approx49\times3.14 = 153.94\space cm^{2}$.

### 4. Word Problems
#### 1. Sale price of a phone originally $\$950$ with $18\%$ off
# Answer:
$\$779$
# Explanation:
## Step1: Find the discount amount
The discount amount is $18\%$ of $950$. Calculate $0.18\times950 = 171$.
## Step2: Subtract the discount from the original price
The sale price is $950-171=\$779$.

#### 2. Distance traveled by a car with speed $62$ mph in $3.5$ hours
# Answer:
$217$ miles
# Explanation:
## Step1: Recall the formula $d=v\times t$ (distance = speed×time)
Here, $v = 62$ mph and $t = 3.5$ hours.
## Step2: Substitute the values
$d=62\times3.5 = 217$ miles.

#### 3. Score needed on the fifth test to have an average of $85$
# Answer:
$75$
# Explanation:
## Step1: Recall the formula for the average
The average $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Let the score on the fifth test be $x$. The sum of the first four scores is $78 + 85+90 + 87=340$. The average of five scores is $\frac{340 + x}{5}=85$.
## Step2: Solve for $x$
Multiply both sides by 5: $340+x=425$. Then subtract 340 from both sides: $x=425 - 340 = 75$.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

1. Solve \(\frac{x}{3}+6 = 10\)

Step1: Subtract 6 from both sides

Step2: Multiply both sides by 3

2. Linear Functions

1. Write the equation of a line with slope \(m = 3\) passing through \((2,5)\)

Step1: Use the point - slope form

Step2: Simplify the equation

2. Determine if the functions are linear

Step1: Recall the form of a linear function

Answer: