Answer :
To graph the line with slope -1 passing through the point [tex]\((5, -3)\)[/tex], we will follow these steps:
### Step 1: Understand the Slope-Intercept Form
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 2: Determine the Equation of the Line
Given:
- Slope [tex]\( m = -1 \)[/tex]
- Point [tex]\((x_1, y_1) = (5, -3)\)[/tex]
We need to find the y-intercept [tex]\( b \)[/tex]. Substitute the point [tex]\((5, -3)\)[/tex] into the equation [tex]\( y = mx + b \)[/tex]:
[tex]\[ -3 = (-1)(5) + b \][/tex]
[tex]\[ -3 = -5 + b \][/tex]
Add 5 to both sides to solve for [tex]\( b \)[/tex]:
[tex]\[ -3 + 5 = b \][/tex]
[tex]\[ b = 2 \][/tex]
So, the equation of the line is:
[tex]\[ y = -x + 2 \][/tex]
### Step 3: Plot the Line
Now, we will plot the line using its equation [tex]\( y = -x + 2 \)[/tex].
#### Step 3.1: Identify the y-Intercept
The y-intercept is [tex]\( b = 2 \)[/tex]. This is where the line crosses the y-axis. The point is [tex]\((0, 2)\)[/tex].
#### Step 3.2: Identify Another Point Using the Slope
Starting from the y-intercept [tex]\((0, 2)\)[/tex] and using the slope [tex]\( m = -1 \)[/tex], move one unit right along the x-axis and one unit down along the y-axis (because the slope is -1).
So, another point on the line is [tex]\((1, 1)\)[/tex].
#### Step 3.3: Plot Additional Points if Needed
To make the line more precise, let's use the given point [tex]\((5, -3)\)[/tex].
### Step 4: Draw the Line
Using the points [tex]\((0, 2)\)[/tex], [tex]\((1, 1)\)[/tex], and [tex]\((5, -3)\)[/tex]:
1. Plot these points on the coordinate plane.
2. Draw a straight line through these points.
### Step 5: Verifying the Line
Ensure that the line follows the equation [tex]\( y = -x + 2 \)[/tex]:
- For [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex] (Point: [tex]\((0, 2)\)[/tex]).
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 1 \)[/tex] (Point: [tex]\((1, 1)\)[/tex]).
- For [tex]\( x = 5 \)[/tex], [tex]\( y = -3 \)[/tex] (Point: [tex]\((5, -3)\)[/tex]).
These calculations confirm that our line is correct.
### Final Plot
On the coordinate plane:
- The line passes through [tex]\((0, 2)\)[/tex], [tex]\((1, 1)\)[/tex], and [tex]\((5, -3)\)[/tex].
- The equation of the line is [tex]\( y = -x + 2 \)[/tex].
You should be able to see that the line runs diagonally from top left to bottom right, crossing the y-axis at 2 and correctly going through the point [tex]\((5, -3)\)[/tex].
### Step 1: Understand the Slope-Intercept Form
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 2: Determine the Equation of the Line
Given:
- Slope [tex]\( m = -1 \)[/tex]
- Point [tex]\((x_1, y_1) = (5, -3)\)[/tex]
We need to find the y-intercept [tex]\( b \)[/tex]. Substitute the point [tex]\((5, -3)\)[/tex] into the equation [tex]\( y = mx + b \)[/tex]:
[tex]\[ -3 = (-1)(5) + b \][/tex]
[tex]\[ -3 = -5 + b \][/tex]
Add 5 to both sides to solve for [tex]\( b \)[/tex]:
[tex]\[ -3 + 5 = b \][/tex]
[tex]\[ b = 2 \][/tex]
So, the equation of the line is:
[tex]\[ y = -x + 2 \][/tex]
### Step 3: Plot the Line
Now, we will plot the line using its equation [tex]\( y = -x + 2 \)[/tex].
#### Step 3.1: Identify the y-Intercept
The y-intercept is [tex]\( b = 2 \)[/tex]. This is where the line crosses the y-axis. The point is [tex]\((0, 2)\)[/tex].
#### Step 3.2: Identify Another Point Using the Slope
Starting from the y-intercept [tex]\((0, 2)\)[/tex] and using the slope [tex]\( m = -1 \)[/tex], move one unit right along the x-axis and one unit down along the y-axis (because the slope is -1).
So, another point on the line is [tex]\((1, 1)\)[/tex].
#### Step 3.3: Plot Additional Points if Needed
To make the line more precise, let's use the given point [tex]\((5, -3)\)[/tex].
### Step 4: Draw the Line
Using the points [tex]\((0, 2)\)[/tex], [tex]\((1, 1)\)[/tex], and [tex]\((5, -3)\)[/tex]:
1. Plot these points on the coordinate plane.
2. Draw a straight line through these points.
### Step 5: Verifying the Line
Ensure that the line follows the equation [tex]\( y = -x + 2 \)[/tex]:
- For [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex] (Point: [tex]\((0, 2)\)[/tex]).
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 1 \)[/tex] (Point: [tex]\((1, 1)\)[/tex]).
- For [tex]\( x = 5 \)[/tex], [tex]\( y = -3 \)[/tex] (Point: [tex]\((5, -3)\)[/tex]).
These calculations confirm that our line is correct.
### Final Plot
On the coordinate plane:
- The line passes through [tex]\((0, 2)\)[/tex], [tex]\((1, 1)\)[/tex], and [tex]\((5, -3)\)[/tex].
- The equation of the line is [tex]\( y = -x + 2 \)[/tex].
You should be able to see that the line runs diagonally from top left to bottom right, crossing the y-axis at 2 and correctly going through the point [tex]\((5, -3)\)[/tex].
