write an equation in slope - intercept form of the line that passes through the given points.
28. (0, 3) and (2, 5)
29. (-2, 4) and (3, -1)
30. (-3, 3) and (1, 2)
graph each equation.
31. $y = x + 5$
32. $y = 3x + 4$
33. $y = -2x + 1$
34. retail sales suppose you have a $5 - off coupon at a fabric store. you buy fabric that costs $7.50 per yard. write an equation that models the total amount of money $y$ you pay if you buy $x$ yards of fabric. what is the graph of the equation?
35. temperature the temperature at sunrise is $65^{circ} ext{f}$. each hour during the day, the temperature rises $5^{circ} ext{f}$. write an equation that models the temperature $y$, in degrees fahrenheit, after $x$ hours during the day. what is the graph of the equation?

Question

write an equation in slope - intercept form of the line that passes through the given points.
28. (0, 3) and (2, 5)
29. (-2, 4) and (3, -1)
30. (-3, 3) and (1, 2)
graph each equation.
31. $y = x + 5$
32. $y = 3x + 4$
33. $y = -2x + 1$
34. retail sales suppose you have a $5 - off coupon at a fabric store. you buy fabric that costs $7.50 per yard. write an equation that models the total amount of money $y$ you pay if you buy $x$ yards of fabric. what is the graph of the equation?
35. temperature the temperature at sunrise is $65^{circ}	ext{f}$. each hour during the day, the temperature rises $5^{circ}	ext{f}$. write an equation that models the temperature $y$, in degrees fahrenheit, after $x$ hours during the day. what is the graph of the equation?

Accepted Answer

### Problem 28:
# Explanation:
## Step 1: Find the slope ($m$)
The formula for slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $(0, 3)$ and $(2, 5)$, $x_1 = 0,y_1 = 3,x_2 = 2,y_2 = 5$. So $m=\frac{5 - 3}{2 - 0}=\frac{2}{2}=1$.
## Step 2: Find the y - intercept ($b$)
The slope - intercept form is $y=mx + b$. The line passes through $(0, 3)$, when $x = 0,y = 3$. Substituting into $y=mx + b$ (with $m = 1$), we get $3=1\times0 + b$, so $b = 3$.
## Step 3: Write the equation
Using $y=mx + b$ with $m = 1$ and $b = 3$, the equation is $y=x + 3$.

# Answer: $y=x + 3$

### Problem 29:
# Explanation:
## Step 1: Find the slope ($m$)
Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ for points $(-2, 4)$ and $(3,-1)$, where $x_1=-2,y_1 = 4,x_2 = 3,y_2=-1$. Then $m=\frac{-1 - 4}{3-(-2)}=\frac{-5}{5}=-1$.
## Step 2: Find the y - intercept ($b$)
Using the slope - intercept form $y=mx + b$. Substitute $m=-1$ and the point $(-2, 4)$ (we can use either point) into the equation: $4=-1\times(-2)+b$, which simplifies to $4 = 2 + b$. Subtract 2 from both sides: $b=4 - 2=2$.
## Step 3: Write the equation
Using $y=mx + b$ with $m=-1$ and $b = 2$, the equation is $y=-x + 2$.

# Answer: $y=-x + 2$

### Problem 30:
# Explanation:
## Step 1: Find the slope ($m$)
For points $(-3, 3)$ and $(1, 2)$, using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$, where $x_1=-3,y_1 = 3,x_2 = 1,y_2 = 2$. So $m=\frac{2 - 3}{1-(-3)}=\frac{-1}{4}=-\frac{1}{4}$.
## Step 2: Find the y - intercept ($b$)
Substitute $m =-\frac{1}{4}$ and the point $(1, 2)$ into $y=mx + b$: $2=-\frac{1}{4}\times1 + b$. Add $\frac{1}{4}$ to both sides: $b=2+\frac{1}{4}=\frac{8 + 1}{4}=\frac{9}{4}$.
## Step 3: Write the equation
Using $y=mx + b$ with $m =-\frac{1}{4}$ and $b=\frac{9}{4}$, the equation is $y=-\frac{1}{4}x+\frac{9}{4}$.

# Answer: $y =-\frac{1}{4}x+\frac{9}{4}$

### Problem 31:
To graph $y=x + 5$:
- **Step 1: Identify the slope and y - intercept**
The slope - intercept form is $y=mx + b$, where $m = 1$ (slope) and $b = 5$ (y - intercept).
- **Step 2: Plot the y - intercept**
Plot the point $(0, 5)$ since when $x = 0,y = 5$.
- **Step 3: Use the slope to find another point**
The slope $m = 1=\frac{1}{1}$, which means from the point $(0, 5)$, move 1 unit up (change in $y$) and 1 unit to the right (change in $x$) to get the point $(1, 6)$.
- **Step 4: Draw the line**
Draw a straight line through the points $(0, 5)$ and $(1, 6)$ (and extend it in both directions).

### Problem 32:
To graph $y = 3x+4$:
- **Step 1: Identify the slope and y - intercept**
Here, $m = 3$ (slope) and $b = 4$ (y - intercept).
- **Step 2: Plot the y - intercept**
Plot the point $(0, 4)$ (since when $x = 0,y = 4$).
- **Step 3: Use the slope to find another point**
The slope $m = 3=\frac{3}{1}$, so from $(0, 4)$, move 3 units up and 1 unit to the right to get the point $(1, 7)$.
- **Step 4: Draw the line**
Draw a straight line through $(0, 4)$ and $(1, 7)$ (and extend it).

### Problem 33:
To graph $y=-2x + 1$:
- **Step 1: Identify the slope and y - intercept**
Here, $m=-2$ (slope) and $b = 1$ (y - intercept).
- **Step 2: Plot the y - intercept**
Plot the point $(0, 1)$ (when $x = 0,y = 1$).
- **Step 3: Use the slope to find another point**
The slope $m=-2=\frac{-2}{1}$, so from $(0, 1)$, move 2 units down (change in $y$) and 1 unit to the right (change in $x$) to get the point $(1,-1)$.
- **Step 4: Draw the line**
Draw a straight line through $(0, 1)$ and $(1,-1)$ (and extend it).

### Problem 34:
# Explanation:
## Step 1: Define the variables and cost components
Let $x$ be the number of yards of fabric and $y$ be the total amount paid. The cost per yard is $\$7.50$, so the cost for $x$ yards is $7.50x$ dollars. We also have a $\$5$ - off coupon, so we subtract 5 from the total cost of the fabric.
## Step 2: Write the equation
The equation is $y = 7.50x-5$.
## Step 3: Analyze the graph
The equation $y = 7.50x-5$ is in slope - intercept form $y=mx + b$ where $m = 7.50$ (slope) and $b=-5$ (y - intercept). Since $x$ represents the number of yards (a non - negative quantity in this context, $x\geq0$), the graph is a line starting from the point $(0,-5)$ (but in the context of the problem, when $x = 0$ (no fabric bought), we would not use the coupon in a practical sense, but mathematically the y - intercept is $-5$) and has a positive slope of $7.50$. For $x\geq0$, we plot points by choosing non - negative values of $x$, calculating $y$ (e.g., when $x = 1$, $y=7.50\times1-5 = 2.50$; when $x = 2$, $y=7.50\times2-5=10$) and drawing a line through these points for $x\geq0$.

# Answer: The equation is $y = 7.50x - 5$. The graph is a line with slope $7.50$ and y - intercept $-5$, defined for $x\geq0$ (practical domain) with points plotted using $x$ (yards) and $y$ (total cost).

### Problem 35:
# Explanation:
## Step 1: Define the variables and temperature change
Let $x$ be the number of hours during the day and $y$ be the temperature in degrees Fahrenheit. The initial temperature (at sunrise, $x = 0$) is $65^{\circ}\text{F}$, and the temperature rises $5^{\circ}\text{F}$ each hour. So the change in temperature per hour is the slope, and the initial temperature is the y - intercept.
## Step 2: Write the equation
Using the slope - intercept form $y=mx + b$, where $m = 5$ (rate of temperature rise) and $b = 65$ (initial temperature), the equation is $y=5x + 65$.
## Step 3: Analyze the graph
The equation $y = 5x+65$ is in slope - intercept form with $m = 5$ (slope) and $b = 65$ (y - intercept). Since $x$ represents the number of hours during the day ($x\geq0$), the graph is a line starting at the point $(0, 65)$ (when $x = 0$, temperature is $65^{\circ}\text{F}$) with a positive slope of 5. We can plot points by choosing non - negative values of $x$ (e.g., when $x = 1$, $y=5\times1 + 65=70$; when $x = 2$, $y=5\times2+65 = 75$) and drawing a line through these points for $x\geq0$.

# Answer: The equation is $y = 5x + 65$. The graph is a line with slope 5 and y - intercept 65, defined for $x\geq0$ (number of hours during the day) with points plotted using $x$ (hours) and $y$ (temperature).

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QUESTION IMAGE

Question

Explanation:

Problem 28:

Step 1: Find the slope ($m$)

Step 2: Find the y - intercept ($b$)

Step 3: Write the equation

Step 1: Find the slope ($m$)

Step 2: Find the y - intercept ($b$)

Step 3: Write the equation

Step 1: Find the slope ($m$)

Step 2: Find the y - intercept ($b$)

Step 3: Write the equation

Answer:

Problem 29: