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# Y MX B Calculator + Online Solver With Free Steps

The **Y MX B Calculator** plots a line and solves for its roots given the slope-intercept form or equation of a line y = mx + b. Here, m represents the slope of the line and b the y-intercept (where the line intersects the y-axis).

The calculator assumes the slope and intercept are already known. Otherwise, if you have a linear equation in two variables, you can re-arrange it to get the equation of a line. Then, you just need to compare the re-arranged form with the standard form to get the values m and b.

## What Is the Y MX B Calculator?

**The Y MX B Calculator is an online tool that uses the slope-intercept form or equation of a line to calculate various properties of that line and plot it on a 2D graph.**

The **calculator interface** consists of two text boxes side-by-side. The first one on the left takes the value of the y-intercept b, and the second box on the right takes the value of the slope m.

If you do not have the values of the slope and the y-intercept, you can get them from the slope-intercept form of a line. Consider the equation:

y = 3x + 2

This equation is already in the slope-intercept form. Now compare it with the general slope-intercept form of a line:

y = mx + b

Then, in this case:

slope = m = 3, y-intercept = b = 2

If your equation can be re-arranged into this form, it represents a line, and you can use the calculator!

## How To Use the Y MX B Calculator?

You can use the **Y MX B Calculator** to plot and find the properties of a line by entering the values of the slope and y-intercept. For example, suppose you want to plot a line with slope m = 1.53 and b = 6.17. You can use the calculator for this by following the step-by-step guidelines below.

### Step 1

Ensure the values for slope and y-intercept do not contain any variables. Otherwise, the shape you are dealing with is probably not a line, and the calculator will not display the plot either.

### Step 2

Enter the value of the y-intercept b into the first text box on the left. In our example’s case, you would type “1.53” without the quotes.

### Step 3

Enter the value of the slope m into the second text box on the right. For this example, you would enter “6.17” without quotes.

### Step 4

Press the **Submit** button to get the results.

### Results

The results span multiple sections, but the most important ones are the **“Plot”** and **“Root”** sections. The former shows the 2D plot of the line and the latter contains the root of the line equation.

Note that this root is essentially the x-intercept of the line – that is, the value of x where y = 0, or visually, the line intersects the x-axis.

There are a few other sections that might be useful:

**Input:**This section contains the input values of the slope and y-intercept plugged into the slope-intercept form of a line for manual verification.**Geometric figure:**The type of figure created by the provided values. If all is well, this should say “line.”**Properties:**This contains the properties of the line as a real function over the variable x. These include the domain, range, and specific properties such as bijectivity.**Partial derivatives:**The partial derivatives of the line equation over x and y, although in the standard form, only the derivative w.r.t. x matters.**Alternate forms:**These are rearranged versions of the slope-intercept line equation.

For our mock example above, the results are:

**Input:** y = 6.17x + 1.53

**Geometric figure: **line

**Root: **-0.247974

**Properties: **Domain $\mathbb{R}$, Range $\mathbb{R}$, bijective

**Partial derivatives: **

$\displaystyle \frac{\partial}{\partial x}$(6.17x + 1.53) = 6.17

$\displaystyle \frac{\partial}{\partial y}$(6.17x + 1.53) = 0

And the plot is given below:

## How Does the Y MX B Calculator Work?

The **Y MX B Calculator **works by plugging the input values for slope m and y-intercept b into the following equation: y = mx + b.

The above equation is the slope-intercept form of a line in two dimensions. The calculator then finds the root of the equation (essentially the x-intercept of the line) by setting y = 0 and solving for x. Finally, it plots it over a range of values for x.

### Slope

The slope or gradient of a 2D line joining two points, or equivalently two points on a line, is the ratio of the difference between their y (vertical) and x (horizontal) coordinates. Thus, the slope represents the sharpness of the rise or fall of the line (y values) compared to the x values.

In other words, a line with a large slope will rise sharply – meaning that, for points on the line, the y component changes much more rapidly than the x component (the line has a large incline).

Similarly, for a line with a small slope, the y component changes much more slowly than the x component (the line has a slight incline).

Sometimes, the definition is shortened to “the ratio of the rise over the run” or just “rise over run,” where **“rise”** is the difference in the vertical coordinate and **“run”** is the difference in the horizontal coordinate.

\[ m = \frac{\text{vertical change}}{\text{horizontal change}} = \frac{\text{rise}}{\text{run}} = \frac{y_2-y_1}{x_2-x_1} = \frac{\Delta y}{\Delta x} \]

Note that the slope-intercept representation of a line cannot represent completely vertical lines as their slope is $\infty$ and consequently undefined. You should use the polar form representation in those cases.

### Intercept

The intercept is a term used to indicate the intersection of a line with one of the coordinate axes. In 2D Cartesian coordinates, these are the x and y-axes, and the corresponding intersections of the line are the x and y-intercept.

Note that the x-intercept is simply the root of the equation representing the line. The y-intercept represents the offset of the line from the origin point. If it is 0, then the line passes through the origin.

The minimum requirements to get the equation of a line are any two points along that line. You can then solve for the slope and intercept yourself (see Example 3).

In other cases, if you have a linear equation in two variables, you can rearrange it to get the slope-intercept form and get the required values from there (see Example 2).

## Solved Examples

### Example 1

Given that a line has a slope of 2 and intersects the y-axis at y = 5, find its slope-intercept form, root(s), and plot it.

### Solution

Given that slope m = 2 and y-intercept b = 5, we simply substitute these values into the standard equation of a line y = mx + b to get the slope-intercept form:

**y = 2x + 5**

If we now put y = 0, we can solve for x to get the root of the equation. Since this is a line, it will only intersect the x-axis at one point and have only one root:

2x + 5 = 0

2x = -5

**x = -2.5**

And plotting this over a range of values of x, we get:

### Example 2

Solve the following equation for y in terms of x.

\[ \sqrt{5x+3y}-3 = 0 \]

### Solution

Isolating the radicals:

\[ \sqrt{5x+3y} = 3 \]

Squaring both sides of the equation:

\[ 5x+3y = 3^2 = 9 \]

Putting all terms on one side:

\[ 5x+3y-9 = 0 \]

It is the equation of a line! Rearranging:

\[ 3y = -5x+9 \]

\[ y = -\frac{5}{3}x + 3 \]

The y-intercept of this line is b = 3, and slope m = -5/3. Setting y = 0, we get the root:

\[ -\frac{5}{3}x + 3 = 0 \, \Rightarrow \, x = \frac{9}{5} \]

**x = 1.8**

Let us plot this:

### Example 3

Consider two points p = (10, 5) and q = (-31, 19). Find the equation of the line joining them and plot it.

### Solution

Let px = 10, py = 5, qx = -31, and qy = 19. Then we can get the slope from the formula:

\[ m = \frac{py – qy}{px – qx} = \frac{5 – 19}{10 – (-31)} \]

\[ m = -\frac{14}{41} \approx -0.341463 \]

Given that p and q are points on the line, we can pick one and the calculated slope value to get the y-intercept value. Let us go with p. Then, putting m = -0.341463, x = px = 10 and y = py = 5 in the equation below:

y = mx + b

b = y – mx

b = 5 – (-0.341463)(10)

b = 5 + 3.41463 = 8.41463

Now that we have both the slope and y-intercept, we can write our line equation as:

y = -0.341463x + 8.41463

And the roots are at y = 0:

-0.341463x + 8.41463 = 0

**x** $\boldsymbol{\approx}$ **24.642875**

Let us further confirm that the point q lies on this line by putting x = qx = -31 and y = qy = 19 in the line equation:

19 = -0.341463(-31) + 8.41463

19 = 10.585353 + 8.41463

19 $\approx$ 18.999983

The slight error above is due to rounding. The line’s plot:

*All graphs/images were created with GeoGebra.*