With this question, it is important to recognize the system of inequalities utilizes the “greater than” sign, not the “greater than or equal to” sign.  does not fulfill .  appears to fulfill both conditions but . 18 is not greater than 8, leaving only  as a possible solution.

With this question, it is important to recognize the system of inequalities utilizes the “greater than or equal to” sign.  does not fulfill wither inequality.  and  fulfill both conditions.

After plotting the lines, shade the regions corresponding to each individual inequality. The area of intersection (in purple) is the solution to the system of inequalities. Another way to check your answer is to pick a point in all four regions delineated by the two equations and test those four coordinate points in the system of inequalities.

This problem tests two key principles related to lines in the coordinate plane. For one, the slopes of perpendicular lines are negative reciprocals of each other (e.g.  and . So here since you need to find a line perpendicular to a line with slope of , you're looking for the negative reciprocal . You can therefore eliminate choices  and , which do not have the proper slope.

The second key concept relates to what a point in the coordinate plane is. Because points are all given in the form  you know that for a point to pass through the point  it will need to allow for  to equal  when  equals . To find the  term (in ), take what you do know about the line:

From here, add  to each side to isolate  and you'll have your answer:

, so the correct equation is .

In order for two lines to be perpendicular, the slopes need to be negative reciprocals of each other. Since the given slope is , you're looking for a slope of .

To find the slope, you need to put the equations in point-slope form, , where  is the slope. Checking the answer choices quickly:

Choices  and  will each give  a positive coefficient, as your first step is to get  on the opposite side of the equation from .

Only choice  simplifies to a coefficient of :

You know that the line goes through point  and the y-intercept of 6 tells you that the line also goes through point . So you can use the "rise over run" slope formula to calculate the slope. Recognizing that the five answer choices focus on , and 6, you should see that you should subtract 0 and 6 (subtracting 0 is the same as doing nothing at all) to get your math looking like the answer choices. So you'll set up the equation:

That Line J passes through the origin gives you a very helpful bit of information regarding point-slope form . That tells you that when , meaning that  meaning that  also equals . That takes a variable away, allowing you to get to work on the two points knowing that the line has a constant slope, :

For point , that means that .

For point , that means that .

This means that you can plug in  into the first equation, allowing you to solve for  (the slope):

 (or ).

When attempting to find where two lines intersect, it is typically best to get the lines in the form . For Line A, you know that the slope is -1, so you have a head start in that . So you're starting with . And then remember - the x-intercept is the point at which , so that point is . You can then plug that point into the equation to find :

, so . You know now that Line A has the equation .

For Line B, you know that the slope is 2 (so ) and that when  is , . Plug that into the line equation to solve for b and you have -2 = 2(0) + b, so b = -2. Now you know the equation for Line B: .

Since  and , the two lines will intersect where . Algebraically that leads you to , so . Plug that back into either line to find y, and you'll find that . Since the point of intersection is , the sum  makes  the correct answer.

We’ll need to use the graph of the linear equation to better understand the existing line, so that we can draw conclusions about the new line. Since the existing line moves vertically by one for every three it moves to the right, our slope, or  is . Since our line crosses the y axis at the point (0, 4), our full equation is .

Thus, our new line will have a slope of  since the question told us that the slope of our new line, , will be twice the slope of the pictured line. Now, to find the full equation of the new line, and thus its y-intercept, we’ll want to plug the known point of the line  into the equation.

If , we can solve for !

, thus .

So, our new line is  (shown below in blue) and its y-intercept is .

To identify the slope of the line, we’ll want to identify two points on the line to find the  otherwise known as the . One point,  is given to us. Our other point can be determined by looking to the y-intercept, in this case, . So, if we identify our  by taking , we arrive at  or , our correct answer.