Find a missing coordinate using slope calculator soup

Define What is the Slope of Line?

The slope of a line in the two-dimensional Cartesian coordinate plane is usually represented by the letter m, and it is sometimes called the rate of change between two points. This is because it is the change in the y-coordinates divided by the corresponding change in the x-coordinates between two distinct points on the line. If we have coordinates of two points `A(x_A,y_A)` and `B(x_B,y_B)` in the two-dimensional Cartesian coordinate plane, then the slope m of the line through `A(x_A,y_A)` and `B(x_B,y_B)` is fully determined by the following formula

`m=\frac{y_B-y_A}{x_B-x_A}`

In other words, the formula for the slope can be written as

$$m=\frac{\Delta y}{\Delta x}=\frac{{\rm vertical \; change}}{{\rm horizontal \; change}}=\frac{{\rm rise}}{{\rm run}}$$

As we know, the Greek letter `∆`, means difference or change. The slope m of a line `y = mx + b` can be defined also as the rise divided by the run. Rise means how high or low we have to move to arrive from the point on the left to the point on the right, so we change the value of `y`. Therefore, the rise is the change in `y`, `∆y`. Run means how far left or right we have to move to arrive from the point on the left to the point on the right, so we change the value of `x`. The run is the change in `x`, `∆x`.

The slope m of a line `y = mx + b` describes its steepness. For instance, a greater slope value indicates a steeper incline. There are four different types of slope:

1. Positive slope `m > 0`, if a line `y = mx + b` is increasing, i.e. if it goes up from left to right
2. Negative slope `m < 0`, if a line `y = mx + b` is decreasing, i.e. if it goes down from left to right
3. Zero slope, `m = 0`, if a line `y = mx + b` is horizonal. In this case, the equation of the line is `y = b`
4. Undefined slope, if a line `y = mx + b` is vertical. This is because division by zero leads to infinities. So, the equation of the line is `x = a`. All vertical lines `x = a` have an infinite or undefined slope.

Real World Problems Using Point Slope of a Line

As we mentioned, the fundamental applications of slope or the rate of change are in geometry, especially in analytic geometry. But, the rate of change is also fundamental to the study of calculus. For non-linear functions, the rate of change varies along the function. The first derivative of the function at a point is the slope of the tangent line to the function at the point. So, the first derivative is the rate of change of the function at the point.

In physics, in definitions of some magnitudes such as displacement, velocity and acceleration, the rate of change play important role. For instance, the rate of change of a function is connected to the average velocity.

The rate of change can be found also in many fields of life, for instance population growth, birth and death rates, etc.

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).

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