Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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].
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
