Standard form equation calculator with one point and slope

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Number Line

The general formula for slope-intercept form is , where represents the slope of the line, and represents the -value of the line’s -intercept.

The slope-intercept form of a linear equation makes it easier for us to identify how steep a line is and where it crosses the -axis.

✨ Drag the points on the graph to see how they affect the equation of the line! ✨

When we're given , we first need to find the slope. Then, we can use the slope and one of the given points to solve for the -value of the -intercept and write the equation in slope-intercept form.

What is Standard Form?

The general formula for the standard form of a linear equation is , where , , and are all integers.

We can go from standard form to slope-intercept form by isolating and simplifying:

What is Point-Slope Form?

The general formula for the point-slope form of a linear equation is , where represents the slope of a line that contains the point (, ).

We can go from point-slope form to slope-intercept form by isolating and simplifying:

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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.

How do you find the equation of a line with one point and the slope calculator?

How do you go from slope form to standard form?