Let's break down how to analyze and graph the given linear equation [tex]\( x - 2y = 5 \)[/tex].
### Step 1: Determine the x-intercept
The x-intercept is the point where the graph of the equation crosses the x-axis. At this point, the value of [tex]\( y \)[/tex] is 0.
To find the x-intercept, we set [tex]\( y = 0 \)[/tex] in the equation:
[tex]\[ x - 2(0) = 5 \][/tex]
[tex]\[ x = 5 \][/tex]
So, the x-intercept is [tex]\( (5, 0) \)[/tex].
### Step 2: Rewrite the equation in slope-intercept form
The slope-intercept form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Starting from the given equation:
[tex]\[ x - 2y = 5 \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ -2y = -x + 5 \][/tex]
Divide by [tex]\(-2\)[/tex]:
[tex]\[ y = \frac{1}{2}x - \frac{5}{2} \][/tex]
From here, we can see that the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex], and the y-intercept [tex]\( b \)[/tex] is [tex]\( -\frac{5}{2} \)[/tex].
### Step 3: Identify the slope
The slope [tex]\( m \)[/tex] of the line is [tex]\( \frac{1}{2} \)[/tex]. This means that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\( \frac{1}{2} \)[/tex] units.
### Graphing the equation
1. Plot the x-intercept: Start by plotting the x-intercept [tex]\( (5, 0) \)[/tex] on the graph.
2. Use the slope: From the point [tex]\( (5, 0) \)[/tex], use the slope to find another point on the graph. Since the slope is [tex]\( \frac{1}{2} \)[/tex], move 2 units to the left (decrease [tex]\( x \)[/tex] by 2), and then move 1 unit down (decrease [tex]\( y \)[/tex] by 1) because [tex]\( -\frac{1}{2} \times 2 = -1 \)[/tex]. This gives us the point [tex]\( (3, -1) \)[/tex].
3. Draw the line: Plot the second point obtained using the slope, [tex]\( (3, -1) \)[/tex]. Draw a straight line through the points [tex]\( (5, 0) \)[/tex] and [tex]\( (3, -1) \)[/tex].
Your graph should look like this:
```
| .
| .
| .
| .
|
(5, 0)
```
This graph accurately represents the equation [tex]\( x - 2y = 5 \)[/tex] with an x-intercept of 5 and a slope of [tex]\( \frac{1}{2} \)[/tex].
