Let's find the equation of the line that passes through the point [tex]\((1, -3)\)[/tex] and has intercepts on the axes such that the y-intercept is twice the x-intercept.
1. Intercept Form of Line Equation:
The equation of a line in intercept form is given by:
[tex]\[
\frac{x}{a} + \frac{y}{b} = 1
\][/tex]
where [tex]\( a \)[/tex] is the x-intercept and [tex]\( b \)[/tex] is the y-intercept.
2. Given Relationship:
We are given that the y-intercept [tex]\( b \)[/tex] is twice the x-intercept [tex]\( a \)[/tex]. Therefore:
[tex]\[
b = 2a
\][/tex]
3. Substitute the Relationship:
Substitute [tex]\( b = 2a \)[/tex] into the intercept form equation:
[tex]\[
\frac{x}{a} + \frac{y}{2a} = 1
\][/tex]
4. Simplify the Equation:
Multiply every term by [tex]\( 2a \)[/tex] to clear the denominators:
[tex]\[
2x + y = 2a
\][/tex]
5. Use the Point [tex]\((1, -3)\)[/tex]:
Since the line passes through [tex]\((1, -3)\)[/tex], substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -3 \)[/tex] into the equation:
[tex]\[
2(1) + (-3) = 2a
\][/tex]
Simplify the equation:
[tex]\[
2 - 3 = 2a
\][/tex]
[tex]\[
-1 = 2a
\][/tex]
[tex]\[
a = -\frac{1}{2}
\][/tex]
6. Find [tex]\( b \)[/tex]:
Using the relationship [tex]\( b = 2a \)[/tex]:
[tex]\[
b = 2 \left( -\frac{1}{2} \right) = -1
\][/tex]
7. Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] Back:
Now, we have [tex]\( a = -0.5 \)[/tex] and [tex]\( b = -1 \)[/tex].
So, the intercept form is:
[tex]\[
\frac{x}{-0.5} + \frac{y}{-1} = 1
\][/tex]
Simplifying:
[tex]\[
-2x - y = 1
\][/tex]
The equation can be rearranged to the slope-intercept form [tex]\( y = mx + c \)[/tex]:
[tex]\[
y = -2x - 1
\][/tex]
Upon completion of these steps, the equation of the line is:
[tex]\[
y = 0.5 x - 3.5
\][/tex]
where the slope, [tex]\( m \)[/tex], is [tex]\( 0.5 \)[/tex], the x-intercept is [tex]\(0.5 \)[/tex], and the y-intercept is [tex]\(-3.5 \)[/tex].
