8) determine the slope and y intercept of the equation $3x + 4y = 12$
9) simplify the expression: $\frac{3y^2 - 12y}{9y - 36}$
10) graph the equation: $y = -2x + 3$
11) determine the slope of a line that contains the points $(1,5)$ and $(9,3)$.
12) write an inequality for the following graphs
a)

b)
13) a hiker walks at a contact speed of 6 miles in 2 hours. what is the rate of change? (label your answer!)

Question

8) determine the slope and y intercept of the equation $3x + 4y = 12$
9) simplify the expression: $\frac{3y^2 - 12y}{9y - 36}$
10) graph the equation: $y = -2x + 3$
11) determine the slope of a line that contains the points $(1,5)$ and $(9,3)$.
12) write an inequality for the following graphs
a)

b)
13) a hiker walks at a contact speed of 6 miles in 2 hours. what is the rate of change? (label your answer!)

Accepted Answer

### Problem 8: Determine the slope and y - intercept of the equation $3x + 4y=12$
# Explanation:
## Step 1: Rewrite in slope - intercept form ($y = mx + b$)
We need to solve the equation $3x + 4y = 12$ for $y$.
Subtract $3x$ from both sides: $4y=-3x + 12$
Then divide each term by 4: $y=-\frac{3}{4}x + 3$
## Step 2: Identify slope and y - intercept
In the slope - intercept form $y = mx + b$, $m$ is the slope and $b$ is the y - intercept.
For $y =-\frac{3}{4}x+3$, the slope $m =-\frac{3}{4}$ and the y - intercept $b = 3$ (or the point $(0,3)$)
# Answer:
Slope: $-\frac{3}{4}$, y - intercept: $3$ (or $(0,3)$)

### Problem 9: Simplify the expression $\frac{3y^{2}-12y}{9y - 36}$
# Explanation:
## Step 1: Factor numerator and denominator
Factor the numerator: $3y^{2}-12y=3y(y - 4)$
Factor the denominator: $9y-36 = 9(y - 4)=3\times3(y - 4)$
So the expression becomes $\frac{3y(y - 4)}{3\times3(y - 4)}$
## Step 2: Cancel common factors
We can cancel out the common factors of $3$ and $(y - 4)$ (assuming $y
eq4$)
$\frac{3y(y - 4)}{3\times3(y - 4)}=\frac{y}{3}$
# Answer:
$\frac{y}{3}$ (for $y
eq4$)

### Problem 10: Graph the equation $y=-2x + 3$
# Explanation:
## Step 1: Identify slope and y - intercept
The equation is in slope - intercept form $y=mx + b$, where $m=-2$ (slope) and $b = 3$ (y - intercept).
## Step 2: Plot the y - intercept
The y - intercept is $(0,3)$. So we plot the point $(0,3)$ on the coordinate plane.
## Step 3: Use the slope to find another point
The slope $m=-2=\frac{-2}{1}$. From the point $(0,3)$, we move down 2 units (because the numerator of the slope is - 2) and then 1 unit to the right (because the denominator of the slope is 1). This gives us the point $(1,1)$. We can also move up 2 units and 1 unit to the left from $(0,3)$ to get the point $(-1,5)$.
## Step 4: Draw the line
Draw a straight line through the points $(0,3)$, $(1,1)$ (or $(-1,5)$) to graph the line $y=-2x + 3$
# Answer:
The graph is a straight line with y - intercept $(0,3)$ and slope - 2. Points on the line include $(0,3)$, $(1,1)$, $(-1,5)$ etc. (graph drawn through these points)

### Problem 11: Determine the slope of a line that contains the points $(1,5)$ and $(9,3)$
# Explanation:
## Step 1: Recall the slope formula
The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$
## Step 2: Substitute the values
Let $(x_1,y_1)=(1,5)$ and $(x_2,y_2)=(9,3)$
Then $m=\frac{3 - 5}{9 - 1}=\frac{-2}{8}=-\frac{1}{4}$
# Answer:
$-\frac{1}{4}$

### Problem 12a: Write an inequality for the graph
# Explanation:
## Step 1: Analyze the graph
The graph has an open circle at $x = 1$ and the arrow is pointing to the left (towards negative infinity). An open circle means that $x = 1$ is not included in the solution set.
## Step 2: Write the inequality
Since the values of $x$ are less than 1 (and 1 is not included), the inequality is $x\lt1$
# Answer:
$x\lt1$

### Problem 12b: Write an inequality for the graph
# Explanation:
## Step 1: Analyze the graph
The graph has a closed circle at $x=-4$ and the arrow is pointing to the right (towards positive infinity). A closed circle means that $x=-4$ is included in the solution set.
## Step 2: Write the inequality
Since the values of $x$ are greater than or equal to - 4, the inequality is $x\geq - 4$
# Answer:
$x\geq - 4$

### Problem 13: A hiker walks at a constant speed of 6 miles in 2 hours. What is the rate of change? (label your answer!)
# Explanation:
## Step 1: Recall the formula for rate of change (speed)
Rate of change (speed) $r=\frac{\text{distance}}{\text{time}}$
## Step 2: Substitute the values
The distance $d = 6$ miles and the time $t = 2$ hours.
So $r=\frac{6}{2}=3$ miles per hour.
# Answer:
3 miles per hour

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 8: Determine the slope and y - intercept of the equation \(3x + 4y=12\)

Step 1: Rewrite in slope - intercept form (\(y = mx + b\))

Step 2: Identify slope and y - intercept

Step 1: Factor numerator and denominator

Step 2: Cancel common factors

Step 1: Identify slope and y - intercept

Step 2: Plot the y - intercept

Step 3: Use the slope to find another point

Step 4: Draw the line

Answer:

Problem 9: Simplify the expression \(\frac{3y^{2}-12y}{9y - 36}\)