Find the missing coordinate using the given slope calculator

2 Answers By Expert Tutors

Kirk S. answered • 10/12/14

Mathematics, BSc + 18 Graduate credits in Mathematics

Given that we have slope m=1/3 and a point on this slope of (-7,-4), we will use the Formula for the slope of a straight line to determine the unknown for the coordinate (x,0) a second point on the slope of this line.

Let (x1,y1) be (-7,-4) and let (x2,y2) be (x,0)

then,

Slope formula    --> m = (y2-y1)/(x2-x1)

--> 1/3 = (0- (-4))/(x-(-7))

--> 1/3 = (0+4)/(x+7)

--> 1/3 = 4/(x+7)

Rationalizing the fractions we will multiply both sides by the least common multiple 3(x+7)

--> x+7 = 12

--> x= 12-7

--> x = 5

Given point (-7,-4) with a slope of 1/3, our answer will reflect the unknown coordinate as (5,0). Where x=5.

The slope of a line is simply rise over run taken from two points, that is take the difference between the y coordinates divided by the difference of the x coordinates. In this case, you got an unknown coordinate being x, and a known slope being 1/3.

Now write an equation for slope and equate it to the given slope in which is:

(0 - -4)/(x--7) = 4/(x+7) = 1/3

Solve for x by using cross multiplying in which is:

x + 7 = 4*3 = 12.

Subtract 7 from both sides in which gives x = 5. Therefore, the unknown coordinate is (5,0).

Being able to find the missing coordinates on a line is often a problem you need to solve to program video games, do well in your algebra class or be proficient in solving coordinate geometry problems. If you want to become an architect, an engineer or a draftsman, you will need to find missing coordinates as part of your job. A common algebra problem requires that you find a missing coordinate (either x or y) given the slope of the line, one pair of known (x, y) coordinates and another (x, y) coordinate pair that has only one known coordinate.

Write down the formula for the slope of the line as M = (Y2 - Y1)/(X2 - X1), where M is the slope of the line, Y2 is the y-coordinate of a point called "A" on the line, X2 is the x-coordinate of point "A," Y1 is the y-coordinate of a point called "B" on the line and X1 is the x-coordinate of point B.

Substitute the value of the slope given and the given coordinate values of point A and point B. Use a slope of "1" and the coordinates of point A as (0, 0) for the point (X2, Y2) and the coordinates of point B as (1, Y1) for the other point (X1, Y1), where Y1 is the unknown coordinate that you must solve for. Check that after you substitute these values into the slope formula that the slope equation reads 1 = (0 - Y1)/(0 - 1).

Solve for the missing coordinate by algebraically manipulating the equation such that the missing coordinate variable is on the left side of the equation and actual coordinate value you must solve for is on the right side of the equation. Use the "Basic Rules of Algebra" link (see Resources) if you are not familiar with solving algebraic equations.

Observe that for this example, the equation, 1 = (0 - Y1)/(0 - 1), simplifies to 1 = -Y1/-1 since subtracting a number from 0 is the negative of the number itself. And so 1 = Y1/1. Conclude that the missing coordinate, Y1, is equal to 1, since, 1 = Y1 is the same as Y1 = 1.

Warnings

• The most common mistake in solving for missing coordinates is not entering the coordinates in the right order when you substitute the coordinates into the slope equation (mixing up the order of X1 and X2 or Y1 and Y2). This will result in a slope that has the wrong sign (a negative slope instead of a positive slope or a positive slope instead of a negative slope).

One of the forms we use when writing linear equations with two variables is the point-slope form. {eq}y - y_1 = m(x - x_1) \\ m \ \text{is the slope.} \\ (x_1, y_1) \ \text{are the coordinates of a point.} \\ {/eq}

In this lesson, we are going to look at two examples.

Example Problem 1: Finding a Missing Coordinate Using A Slope and a Point

A line with a slope of 3 passes through (2, 1).

(a) Write an equation of the line in slope-intercept form.

(b) Find the y-coordinate of a point on the line whose x-coordinate is -1.

Solution:

(a) Since the slope is 3, then {eq}m = 3. {/eq}

We are given the coordinates of a point, so:

{eq}x_1 = 2 \\ y_1 = 1 {/eq}

Now, substitute all the values into the point-slope formula:

{eq}y - 1 =3(x - 2) {/eq}

Simplify and solve for y:

{eq}y - 1 =3x - 6 \\ y - 1 + 1 = 3x - 6 + 1 \\ y = 3x - 5 {/eq}

(b) To find the y-coordinate, we will substitute -1 for x:

{eq}y = 3 \cdot (-1) - 5 \\ y = -3 - 5 \\ y = -8 \\ {/eq}

Example Problem 2: Finding a Missing Coordinate Given Two Points

A line passes through points (-2, 4) and (-4, 3)

(a) Write an equation of the line in the slope-intercept form.

(b) Find the y-coordinate of a point on the line whose x-coordinate is 6.

Solution:

(a) Our first step is to find the slope of the line, which is given by the following relation, for two known points on the line:

{eq}m = \dfrac{y_2 - y_1}{x_2 - x_1} \\ (x_1, y_1) = (-2, 4) \\ (x_2, y_2) = (-4, 3) \\[0.4 cm] m = \dfrac{3 - 4}{-4 - (-2)} \\ m = \dfrac{-1}{-2} \\ m = \dfrac{1}{2} \\ {/eq}

Now, the process is similar to what we did in Example 1:

{eq}m = \frac{1}{2} \\ x_1 = -2 \\ y_1 = 4 \\[0.4 cm] y - 4 = \dfrac{1}{2}(x - (-2)) \\ y - 4 = \dfrac{1}{2}(x + 2) \\ y - 4 = \dfrac{1}{2}x + \dfrac{1}{2} \cdot 2 \\ y - 4 = \dfrac{1}{2}x + 1 \\ y - 4 + 4 = \dfrac{1}{2}x + 1 + 4 \\ y = \dfrac{1}{2}x + 5 \\ {/eq}

(b) Substitute 6 for x:

{eq}y = \dfrac{1}{2} \cdot 6 + 5 \\ y = 3 + 5 \\ y = 8 \\ {/eq}

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