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1.08 Writing Equations of Lines Practice Assignment

1. Write the equation of the line that contains the points (5, -7) and (-1, 5).
Equation of the line: __________
Then graph the line.

2. A certain campground charges a one-time entry fee and a daily cabin rental fee.
a. If the total cost for a 3-day stay is [tex]$90 and for a 5-day stay is $[/tex]130, write the system of equations to represent this situation.
b. Solve the system of equations to find the entry fee and daily rental fee.

Sagot :

GinnyAnswer
Let's solve the problem step-by-step. We need to find the equation of the line that passes through the points (5, -7) and (-1, 5). The equation of a line in slope-intercept form is given by:

[tex]\[ y = mx + b \][/tex]

where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.

### Step 1: Calculate the Slope [tex]\( m \)[/tex]

The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:

[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Let's substitute the given points [tex]\((5, -7)\)[/tex] and [tex]\((-1, 5)\)[/tex]:

[tex]\[ m = \frac{5 - (-7)}{-1 - 5} \][/tex]
[tex]\[ m = \frac{5 + 7}{-1 - 5} \][/tex]
[tex]\[ m = \frac{12}{-6} \][/tex]
[tex]\[ m = -2.0 \][/tex]

### Step 2: Calculate the Y-Intercept [tex]\( b \)[/tex]

The y-intercept [tex]\( b \)[/tex] can be found using the slope-intercept form [tex]\( y = mx + b \)[/tex] by substituting one of the points and the slope. We'll use the point [tex]\((5, -7)\)[/tex]:

[tex]\[ -7 = (-2.0)(5) + b \][/tex]
[tex]\[ -7 = -10 + b \][/tex]
[tex]\[ b = -7 + 10 \][/tex]
[tex]\[ b = 3.0 \][/tex]

### Step 3: Write the Equation of the Line

Now that we have the slope [tex]\( m = -2.0 \)[/tex] and the y-intercept [tex]\( b = 3.0 \)[/tex], we can write the equation of the line:

[tex]\[ y = -2.0x + 3.0 \][/tex]

### Summary

The equation of the line passing through the points (5, -7) and (-1, 5) is:

[tex]\[ y = -2.0x + 3.0 \][/tex]

### Graphing the Line

1. Plot the Points: Start by plotting the given points (5, -7) and (-1, 5) on the coordinate plane.

2. Draw the Line: Use a ruler to draw a straight line through these two points. This line represents the equation [tex]\( y = -2.0x + 3.0 \)[/tex].

To assist with graphing:
- The y-intercept [tex]\( b = 3.0 \)[/tex] means the line crosses the y-axis at (0, 3).
- The slope [tex]\( m = -2.0 \)[/tex] indicates that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 2 units.

This will help ensure the line is drawn accurately on the graph.
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