# determining the equation of a line given slope and two intersecting lines? [closed]

a line passes through the point of intersection of the lines $y = -\frac12x - 6$ and $y = 2x + 4$. determine the equation of the line if it has a slope of $\frac12$.

I'm completely lost, how should i answer this? I know that the line that has a slope of $\frac12$ is intersecting through $y = -\frac12x - 6$ and $y = 2x + 4$, which means that the points to these intersecting lines are the same to the line i'm trying to find the equation for, so how should i figure out those points? (unless my method is wrong)

## closed as off-topic by 5xum, user8795, Namaste, Henrik, GlorfindelAug 16 '17 at 20:39

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – 5xum, Namaste, Henrik, Glorfindel
If this question can be reworded to fit the rules in the help center, please edit the question.

OK, you are looking for a line, so you are looking for $k$ and $n$ in the equation $y=kx+n$. Basically, you want to calculate $k$ and $n$.

Hint:

1. What is the slope of the line $y=kx+n$?
2. What is the intersection of the lines $y=-\frac12 x - 6$ and $y=2x+4$?
3. If I know that the point $(x_0,y_0)$ is on the line $y=kx+n$, what equation do $x_0$ and $y_0$ satisfy?
• 1) 1/2, 2) L1 (0, -6) L2 (0,4), so do i use one of these points to satisfy the equation and solve for n? – Jenny B Aug 2 '17 at 6:45
• @ineedgr10mathhelp wrong and wrong. For 1) I am asking in general, if I give you an equation $y=kx+n$, what is the slope of this line? Like, what is the slope of the line $y=2x+5$? And what is the slope of $y=x+1$? For 2) no. Those are the intersections of the two lines with the $y$ axis, but not the intersection of the two lines among themselves. You want to find a point that is on both lines! – 5xum Aug 2 '17 at 6:51

Based on the hints @5xum gave you, you should see a connection between y=kx+n and y=mx+b (which is what most teachers use).

1. The slope of y=kx+n is k. The question gives that to you.

2. You can find the intersection of the two lines algebraically. Since y=y, then -1/2 x - 6 = 2x + 4. Now solve for x. I personally prefer to graph the lines on Desmos.

3. When you find the point, plug in what you know to y=kx+n. So far, you know k (slope) and x&y (the point of intersection). Remember to multiply k times x. You are left with n as the unknown so solve for n. (Get the n by itself on one side of the equation and a number on the other side.)

4. For your final equation, set up y=kx+n and fill in the k (slope) and n (y-intercept of the new line).

• Most teachers where I'm from use $y=kx+n$, not $y=mx + b$... – 5xum Aug 2 '17 at 6:57
• I suppose it could be a regional difference. I have occasionally seen y=ax+b – MrsSnider Aug 2 '17 at 7:00

The intersection point of the lines $y = -\frac12x-6$ and $y = 2x+4$ is a point where the value of $y$ for both the lines is same. So, $$-\frac12x-6 = 2x+4$$ $$-\frac52x = 10$$ $$x = -4$$ So the point of intersection is at $x=-4$ and $y = -\frac12x-6 = -\frac12(-4) - 6 = -4$. The point of intersection is $(-4, -4)$. Now we know that the unknown line with slope $\frac12$ passes through the point $(-4, -4)$.

I hope this helps.

Any line passing through the intersection of the given lines is

$$\lambda( -x/2-6) +(1-\lambda)(2 x +4) = x/2 + C$$

Put $x=0$ in the above equation,it gives

$-10 \lambda +4 = C \tag1$

Collecting $x$ term coefficients and equating

$$-\lambda/2 + 2 - 2\lambda = \frac12 ,\, \lambda =\frac35 \tag2$$

Eliminate $\lambda$ from (1), (2)

$$C= -2, \tag3$$

So required straight line has equation

$$y = x/2 -2.$$