Answer :
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]
