To find the [tex]\( y \)[/tex]-intercept of the line passing through points [tex]\( M(-3, 5) \)[/tex] and [tex]\( N(2, 0) \)[/tex], we first need to determine the equation of the line. Here's the step-by-step process:
### Step 1: Identify the slope
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For points [tex]\( M(-3, 5) \)[/tex] and [tex]\( N(2, 0) \)[/tex]:
[tex]\[
m = \frac{0 - 5}{2 - (-3)} = \frac{-5}{2 + 3} = \frac{-5}{5} = -1
\][/tex]
### Step 2: Write the equation in point-slope form
Using the point-slope form of a line equation, [tex]\( y - y_1 = m(x - x_1) \)[/tex], and choosing point [tex]\( M(-3, 5) \)[/tex], we get:
[tex]\[
y - 5 = -1(x + 3)
\][/tex]
### Step 3: Simplify the equation and isolate the y variable
Start by distributing the slope [tex]\( -1 \)[/tex] on the right side:
[tex]\[
y - 5 = -x - 3
\][/tex]
Next, isolate [tex]\( y \)[/tex] by adding 5 to both sides:
[tex]\[
y = -x - 3 + 5
\][/tex]
Simplifying this gives:
[tex]\[
y = -x + 2
\][/tex]
From the equation [tex]\( y = -x + 2 \)[/tex], we can see that the [tex]\( y \)[/tex]-intercept (the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]) is [tex]\( 2 \)[/tex].
Thus, the [tex]\( y \)[/tex]-intercept of the line passing through points [tex]\( M \)[/tex] and [tex]\( N \)[/tex] is [tex]\( 2 \)[/tex].
### Standard Form of the Line Equation
The standard form of a line's equation is [tex]\( Ax + By = C \)[/tex]. To convert our equation [tex]\( y = -x + 2 \)[/tex] into standard form, we simply rearrange it:
[tex]\[
y = -x + 2 \implies x + y = 2
\][/tex]
So, the equation of the line in standard form is:
[tex]\[
x + y = 2
\][/tex]
