Question

what is the equation of a line that goes through the point (-6, 0) and has a slope of -1?
hint: use the formula sheet to determine the formula(s) needed to solve the problem.

a y = -x + 6

b y = -x - 6

c y = -x

d y = -6x - 6

Explanation:

Step1: Recall point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope of the line.
We are given that the point $(x_1,y_1)=(-6,0)$ and the slope $m = - 1$.
Substitute these values into the point - slope form:
$y-0=-1(x - (-6))$

Step2: Simplify the equation

Simplify the right - hand side of the equation:
$y=-1(x + 6)$
Using the distributive property $a(b + c)=ab+ac$, where $a=-1$, $b = x$ and $c = 6$, we get:
$y=-x-6$? Wait, no, wait. Wait, $y-0=-1(x - (-6))$ is $y=-1(x + 6)=-x - 6$? Wait, no, let's check the point $(-6,0)$ in the equation $y=-x - 6$.
When $x=-6$, $y=-(-6)-6=6 - 6=0$. Wait, but let's check option A: $y=-x + 6$. When $x=-6$, $y=-(-6)+6=6 + 6 = 12
eq0$. Option B: $y=-x-6$, when $x = - 6$, $y=-(-6)-6=6 - 6=0$. Wait, but wait, let's re - do the point - slope form.
Wait, the point - slope form is $y - y_1=m(x - x_1)$. So $x_1=-6$, $y_1 = 0$, $m=-1$.
So $y-0=-1(x-(-6))\Rightarrow y=-1(x + 6)\Rightarrow y=-x-6$? But when we plug $x=-6$ into $y=-x - 6$, we get $y = 6-6 = 0$, which is correct. But wait, let's check option A again. Wait, maybe I made a mistake. Wait, $y=-x + 6$, when $x=-6$, $y=6 + 6=12
eq0$. Option B: $y=-x-6$, $x=-6$, $y=6 - 6=0$. Option C: $y=-x$, $x=-6$, $y = 6
eq0$. Option D: $y=-6x-6$, $x=-6$, $y=-6\times(-6)-6=36 - 6 = 30
eq0$. Wait, but wait, the correct equation should be $y=-x-6$? But let's re - derive.
Wait, the slope - intercept form is $y=mx + b$. We know $m=-1$, so $y=-x + b$. We know the line passes through $(-6,0)$, so substitute $x=-6$ and $y = 0$ into $y=-x + b$:
$0=-(-6)+b\Rightarrow0 = 6 + b\Rightarrow b=-6$. So the equation is $y=-x-6$? But that's option B. Wait, but wait, the original calculation:
Wait, $y - y_1=m(x - x_1)$: $y-0=-1(x-(-6))\Rightarrow y=-1(x + 6)\Rightarrow y=-x-6$. Yes. So the correct equation is $y=-x-6$? But wait, let's check the point $(-6,0)$ in $y=-x-6$: $y=-(-6)-6=6 - 6 = 0$, correct. In option A: $y=-x + 6$, $x=-6$, $y=6 + 6=12
eq0$. So the correct answer is B? Wait, no, wait, I think I made a mistake. Wait, $y=-x + 6$: when $x = 0$, $y = 6$; when $y = 0$, $x = 6$. The line $y=-x-6$: when $x = 0$, $y=-6$; when $y = 0$, $x=-6$. So the line with slope - 1 passing through $(-6,0)$ is $y=-x-6$? But let's re - do the slope - intercept form.
We know that the slope $m=-1$, so the equation is $y=-x + b$. Substitute $x=-6$, $y = 0$:
$0=-(-6)+b\Rightarrow0 = 6 + b\Rightarrow b=-6$. So $y=-x-6$, which is option B. Wait, but the initial calculation when I thought I made a mistake, but actually, the correct equation is $y=-x-6$, which is option B. Wait, but let's check again. Wait, the point - slope formula: $y - y_1=m(x - x_1)$. So $y-0=-1(x-(-6))\Rightarrow y=-x - 6$. Yes, that's correct. So the answer is B.

Answer:

B. $y = -x - 6$