Graphing A Linear Equation; using slope and a point to write an equation in y = mx + b form

Question

A line in the xy - plane passes through the origin and has a slope of 1/7. Which of the following points lies on the line?

A) (0, 7) 

B) (1, 7) 

C) (7, 7) 

D) (14,2)

Answer 1

To answer this question, first let's note what the question tells us:

1. The line passes through the origin, (0,0).
2. The line has a slope of 1/7, meaning that for every increase of 1 in the y direction, x has to increase by 7 (using the rise over run method, 1 rise of y for every 7 run of x).

Thinking about point 2, x is increasing much more quickly than y since x has to go up by 7 for y to go up by just 1. Therefore, since we're starting at the origin where x and y are both 0, the points on this line are going to have much larger x values than y values. Let's take a look at the choices given:

(x, y)

A) (0, 7) 

B) (1, 7) 

C) (7, 7) 

D) (14,2)

Taking a look at the options A through C, notice how all of those options have x values that are either smaller than or equal to their y values. As we saw above, that wouldn't make sense for a point on the line given. Therefore, D is the correct answer.

That might be the quickest way to solve this question without plugging in numbers, which is fine for tests like the SAT but not for an exam that isn't multiple choice. Let's do it the typical way then with the form y = mx + b:

y = mx + b, where m = slope and b = y-intercept

y = 1/7 * x + 0 (notice that b is 0 because the y-intercept, aka the y value when x is 0, is at the origin where x and y are both 0)

y = 1/7 * x

Now you can plug each of the answer choices in for x and y. For example:

A) (0, 7)

y = 1/7 * x

7 = 1/7 * 0

7 = 0

FALSE

But if we look at choice D:

D) (14, 2)

y = 1/7 * x

2 = 1/7 * 14

2 = 2

TRUE

Answer: D

Answer 2

"passes through the origin" means the point is (0,0)

slope is "rise over run" ... that is, (change in y) / (change in x) .. for slope=1/7, y increases 1 for an increase of 7 for x

In y = mx+b (slope-intercept) form, this is y = (1/7)x

The point (14,2) is on the line because 2 = (1/7)*14

Answer 3

The line

y = (1/7)x

Choice D) (14, 2) lies on the line

For y= mx + b

If the line passes through the origin, then it passes through the point (0, 0) which makes b = 0

The line is (1/7)x

In Slope Intercept Form since b = 0 because the line passes through the origin,

y = mx + b = mx + 0 = mx

y = mx which is just the slope 1/7 times x

Checking in Point Slope Form and using (0, 0) as (x₁, y₁)

(y- y₁) = m(x - x₁)

(2 - 0) = (1/7)(14 - 0)

2 = (1/7)(14

2 = 2

You can check the other points to confirm.

I hope this helps

y = (1/7)x

Answer 4

The equation of the line in slope-intercept form is y = 1/7x + 0, or just y = 1/7x.

You can plug each answer choice (x,y) into the equation, and see which one gives you a true statement.

Choice D (14,2) substituting 14 for x, and 2 for y gives you 2 = (1/7)(14); which is 2 = 2.

Choice D is the point that is on the line

Answer 5

To solve algebraically:

Step 1) Find the equation of the line in slope - intercept form.

Because m = 1/7 and b = 0, our equation is y = 1/7x.

Step 2) Substitute each given (x,y) value into our equation to see which one works.

(0,7) y = 1/7x

7 = 1/7(0)

7 = 0 ?

7 DOES NOT equal 0, and so the answer is not (0,7).

(7,7) y = 1/7x

7 = 1/7(7)

7 = 1 ?

7 DOES NOT equal 1, and so the answer is not (7,7)

(1,7) y = 1/7x

7 = 1/7(1)

7 = 1/7 ?

7 DOES NOT equal 1/7, and so the answer is not (1,7).

(14,2) y = 1/7x

2 = 1/7(14)

2 = 2?

2 DOES EQUAL 2, so the answer is (14,2).

To solve graphically:

Step 1: Graph the equation and plot the points given in the answer choices.

Start by graphing (0,0). Then use the slope to figure out where your next point is. Rise (go up) by 1. Run (go right) by 7 to arrive at the second point. Connect the two points to create your line. The put the answers on the graph.

Step 2: Decide which point is on your line.

You will see when you examine the graph that the last point does lie on your line. See the video for more detail !