Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
Sure! Let's work through the steps to find the equation of the line [tex]\(MN\)[/tex] formed by the points [tex]\(M(-3, 5)\)[/tex] and [tex]\(N(2, 0)\)[/tex].
### Step 1: Identify the slope
To find the slope [tex]\(m\)[/tex] of the line passing through the points [tex]\(M(-3, 5)\)[/tex] and [tex]\(N(2, 0)\)[/tex], we use the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of the points:
[tex]\[ x_1 = -3, \quad y_1 = 5, \quad x_2 = 2, \quad y_2 = 0 \][/tex]
[tex]\[ m = \frac{0 - 5}{2 - (-3)} \][/tex]
[tex]\[ m = \frac{-5}{5} \][/tex]
[tex]\[ m = -1 \][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(-1\)[/tex].
### Step 2: Write the equation in point-slope form
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using point [tex]\(M(-3, 5)\)[/tex] and the slope [tex]\(m = -1\)[/tex]:
[tex]\[ y - 5 = -1(x - (-3)) \][/tex]
[tex]\[ y - 5 = -1(x + 3) \][/tex]
So, the equation in point-slope form is:
[tex]\[ y - 5 = -1(x + 3) \][/tex]
### Step 3: Simplify the equation and isolate the [tex]\(y\)[/tex] variable
From the point-slope form equation:
[tex]\[ y - 5 = -1(x + 3) \][/tex]
Distribute the [tex]\(-1\)[/tex]:
[tex]\[ y - 5 = -x - 3 \][/tex]
Add [tex]\(5\)[/tex] to both sides:
[tex]\[ y = -x - 3 + 5 \][/tex]
[tex]\[ y = -x + 2 \][/tex]
So, the equation of the line in slope-intercept form is:
[tex]\[ y = -x + 2 \][/tex]
### [tex]\(y\)[/tex]-intercept of the line [tex]\(MN\)[/tex]
The [tex]\(y\)[/tex]-intercept is the constant term [tex]\(c\)[/tex] in the slope-intercept form [tex]\(y = mx + c\)[/tex]. Thus, the [tex]\(y\)[/tex]-intercept of the line [tex]\(y = -x + 2\)[/tex] is:
[tex]\[ 2 \][/tex]
### Write the equation in standard form
The standard form of a linear equation is:
[tex]\[ Ax + By = C \][/tex]
Starting with the equation [tex]\(y = -x + 2\)[/tex]:
[tex]\[ y = -x + 2 \][/tex]
Add [tex]\(x\)[/tex] to both sides to get:
[tex]\[ x + y = 2 \][/tex]
So, the equation in standard form is:
[tex]\[ 1x + 1y = 2 \][/tex]
In summary:
- The slope [tex]\(m = -1\)[/tex]
- The equation in point-slope form is [tex]\( y - 5 = -1(x + 3) \)[/tex]
- The equation in slope-intercept form is [tex]\( y = -x + 2 \)[/tex]
- The [tex]\(y\)[/tex]-intercept is [tex]\( 2 \)[/tex]
- The equation in standard form is [tex]\( 1x + 1y = 2 \)[/tex]
### Step 1: Identify the slope
To find the slope [tex]\(m\)[/tex] of the line passing through the points [tex]\(M(-3, 5)\)[/tex] and [tex]\(N(2, 0)\)[/tex], we use the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of the points:
[tex]\[ x_1 = -3, \quad y_1 = 5, \quad x_2 = 2, \quad y_2 = 0 \][/tex]
[tex]\[ m = \frac{0 - 5}{2 - (-3)} \][/tex]
[tex]\[ m = \frac{-5}{5} \][/tex]
[tex]\[ m = -1 \][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(-1\)[/tex].
### Step 2: Write the equation in point-slope form
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using point [tex]\(M(-3, 5)\)[/tex] and the slope [tex]\(m = -1\)[/tex]:
[tex]\[ y - 5 = -1(x - (-3)) \][/tex]
[tex]\[ y - 5 = -1(x + 3) \][/tex]
So, the equation in point-slope form is:
[tex]\[ y - 5 = -1(x + 3) \][/tex]
### Step 3: Simplify the equation and isolate the [tex]\(y\)[/tex] variable
From the point-slope form equation:
[tex]\[ y - 5 = -1(x + 3) \][/tex]
Distribute the [tex]\(-1\)[/tex]:
[tex]\[ y - 5 = -x - 3 \][/tex]
Add [tex]\(5\)[/tex] to both sides:
[tex]\[ y = -x - 3 + 5 \][/tex]
[tex]\[ y = -x + 2 \][/tex]
So, the equation of the line in slope-intercept form is:
[tex]\[ y = -x + 2 \][/tex]
### [tex]\(y\)[/tex]-intercept of the line [tex]\(MN\)[/tex]
The [tex]\(y\)[/tex]-intercept is the constant term [tex]\(c\)[/tex] in the slope-intercept form [tex]\(y = mx + c\)[/tex]. Thus, the [tex]\(y\)[/tex]-intercept of the line [tex]\(y = -x + 2\)[/tex] is:
[tex]\[ 2 \][/tex]
### Write the equation in standard form
The standard form of a linear equation is:
[tex]\[ Ax + By = C \][/tex]
Starting with the equation [tex]\(y = -x + 2\)[/tex]:
[tex]\[ y = -x + 2 \][/tex]
Add [tex]\(x\)[/tex] to both sides to get:
[tex]\[ x + y = 2 \][/tex]
So, the equation in standard form is:
[tex]\[ 1x + 1y = 2 \][/tex]
In summary:
- The slope [tex]\(m = -1\)[/tex]
- The equation in point-slope form is [tex]\( y - 5 = -1(x + 3) \)[/tex]
- The equation in slope-intercept form is [tex]\( y = -x + 2 \)[/tex]
- The [tex]\(y\)[/tex]-intercept is [tex]\( 2 \)[/tex]
- The equation in standard form is [tex]\( 1x + 1y = 2 \)[/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
