Coordinate Geometry - GRE Quantitative Reasoning

If we rearrange the second equation it is the same as the first equation. They are the same line.

We should put this equation in the form of y = mx + b, where m is the slope.

We start with 4_x_ + 3_y_ = 7.

Isolate the y term: 3_y_ = 7 – 4_x_

Divide by 3: y = 7/3 – 4/3 * x

Rearrange terms: y = –4/3 * x + 7/3, so the slope is –4/3.

The equation of a line is:

y = mx + b, where m is the slope

4_x_ – 16_y_ = 24

–16_y_ = –4_x_ + 24

y = (–4_x_)/(–16) + 24/(–16)

y = (1/4)x – 1.5

Slope = 1/4

First write the equation in slope intercept form. Add to both sides to get . Now divide both sides by to get . The slope of this line is , so any line that also has a slope of would be parallel to it. The correct answer is .

To be totally clear, solve both lines in slope-intercept form:

2x – 4y = 33; –4y = 33 – 2x; y = –33/4 + 0.5x

2x + 4y = 33; 4y = 33 – 2x; y = 33/4 – 0.5x

These lines are definitely not the same. Nor are they parallel—their slopes differ. Likewise, they cannot be perpendicular (which would require not only opposite slope signs but also reciprocal slopes); therefore, they are non-perpendicular intersecting.

Parallel lines will always have equal slopes. The slope can be found quickly by observing the equation in slope-intercept form and seeing which number falls in the "" spot in the linear equation ,

We are looking for an answer choice in which both equations have the same value. Both lines in the correct answer have a slope of 2, therefore they are parallel.

Lines are parallel when their slopes are the same.

First, we need to place the given equation in the slope-intercept form.

Because the given line has the slope of , the line parallel to it must also have the same slope.

There are several ways to solve this problem. You could solve all of the equations for . This would give you equations in the form . All of the lines with the same value would be parallel. Otherwise, you could figure out the ratio of to when both values are on the same side of the equation. This would suffice for determining the relationship between the two. We will take the first path, though, as this is most likely to be familiar to you.

Let's solve each for :

Here, you need to be a bit more manipulative with your equation. Multiply the numerator and denominator of the value by :

Therefore, , , and all have slopes of

Now, notice that the slope of the line that you have been given is . You know this because slope is merely:

However, for your points, there is no rise at all. You do not even need to compute the value. You know it will be . All lines with slope are of the form , where is the value that has for all points. Based on our data, this is , for is always —no matter what is the value for . So, the parallel answer choice is , as both have slopes of .

To begin, solve your equation for . This will put it into slope-intercept form, which will easily make the slope apparent. (Remember, slope-intercept form is , where is the slope.)

Divide both sides by and you get:

Therefore, the slope is . Now, you need to test your points to see which set of points has a slope of . Remember, for two points and , you find the slope by using the equation:

For our question, the pair and gives us a slope of :

Since the lines are parallel, the slopes must be the same. Therefore, (p+3) divided by (_–2–_5) must equal _–_2. 11 is the only choice that makes that equation true. This can be solved by setting up the equation and solving for p, or by plugging in the other answer choices for p.

Start by converting the original equation to slop-intercept form.

The slope of this line is . A parallel line will have the same slope. Now that we know the slope of our new line, we can use slope-intercept form and the given point to solve for the y-intercept.

Plug the y-intercept into the slope-intercept equation to get the final answer.

Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.

3x + 4y = 12

4y = _–_3x + 12

y = –(3/4)x + 3

slope = _–_3/4

We know that the second line will also have a slope of _–_3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.

y = mx + b

2 = _–_3/4(1) + b

2 = _–_3/4 + b

b = 2 + 3/4 = 2.75

Plug the y-intercept back into the equation to get our final answer.

y = –(3/4)x + 2.75

(4,2) and (15,-4)

All that we really need to ascertain is the slope of our line. So long as a given answer has this slope, it will not matter what its y-intercept is (given the openness of our question). To find the slope, use the formula: m = rise / run = (y1 - y2) / (x1 - x2):

(2 - (-4)) / (4 - 15) = (2 + 4) / -11 = -6/11

Given this slope, our answer is: y = -6/11x + 57.4

Find the slope of the given line: (slope intercept form)

therefore the slope is

Parallel lines have the same slope, so now we need to find the equation of a line with slope and going through point by substituting values into the point-slope formula.

So,

Thus, the new equation is

By using the complementary and supplementary rules of geometry (due to lines m and n being parallel), as well as the fact that the sum of all angles within a triangle is 180, we can carry through the operations through stepwise subtraction of 180.

x = 125 → angle directly below also = 125. Since a line is 180 degrees, 180 – 125 = 55. Since right triangle, 90 + 55 = 145 → rightmost angle of triangle 180 – 145 = 35 which is equal to the reflected angle. Use supplementary rule again for 180 – 35 = 145 = y.

Once can also recognize that both a straight line and triangle must sum up to 180 degrees to skip the last step.

To solve, we will need to find the slope of the line. We know that it is parallel to the line given by the equation, meaning that the two lines will have equal slopes. Find the slope of the given line by converting the equation to slope-intercept form.

The slope of the line will be . In slope intercept-form, we know that the line will be . Now we can use the given point to find the y-intercept.

The final equation for the line will be .

The line parallel to must have a slope of , giving us the equation . To solve for b, we can substitute the values for y and x.

Therefore, the equation of the line is .

Converting the given line to slope-intercept form we get the following equation:

For parallel lines, the slopes must be equal, so the slope of the new line must also be . We can plug the new slope and the given point into the slope-intercept form to solve for the y-intercept of the new line.

Use the y-intercept in the slope-intercept equation to find the final answer.