# Intersection Calculator + Online Solver With Free Steps

The **Intersection Calculator** is used to calculate the intersection point between two lines. The **two lines** are the linear equations with degree 1. The calculator computes the x and y coordinates of the intersecting point in a 2-D plane.

The calculator takes the **linear equations** for the two lines as input and outputs the **intersecting** **point** or the solution of both lines. The two equations are the function of x and y.

If the variable z is entered in one or both of the two equations, the calculator computes only the x-coordinate of the intersecting point and **gives another equation** which is a function of y and z.

The three-variable equation requires **three equations** to compute the complete coordinates of the intersection point. The two equations are not enough for the calculator to compute the numerical values of x, y, and z coordinates of the intersection point.

So, the calculator gives the **numerical values** for the intersection point only for two-variable equations.

## What Is an Intersection Calculator?

**The Intersection Calculator is an online tool that is used to calculate the intersection point of two linear equations or lines in a 2-D plane.**

The **intersection point** is the point where the two lines meet or cross each other, giving the x and y coordinates.

So the intersecting point is the **common point** (x,y) between the two lines. At this point, the x-coordinate and y-coordinate for both the lines are the same.

## How To Use the Intersection Calculator

The Intersection Calculator can be used by following the steps given below:

### Step 1

First, the user enters the **first linear equation** of the two equations in the input block against the title, **Intersection of**. The linear equation is a two-variable equation.

The calculator shows the first equation by **default** as follows:

**y = 3x + 2 **

The default variables used are x and y. The equation is a function of y in terms of x.

The **two variables** can be any alphabet such as (a,b) depending upon the user’s requirement.

### Step 2

Enter the **second linear equation** in the second input tab of the Intersection Calculator. It is entered in the block titled against **and**. The user should use the same two variables as used for the first linear equation for correct results.

The second linear equation set by **default** by the calculator is:

**y = 2x – 1 **

If a **third variable** is entered in any of the two equations, the calculator gives the value for a single coordinate such as x and gives another equation in the result window.

This calculator does not support the 3-D system.

### Step 3

After entering both the equations, the user should press **Submit **button for the calculator to compute the intersection point. If the user forgets to enter one of the two equations, the calculator displays **Not a valid input; please try again**.

### Output

The calculator processes the two equations and shows the output in the two windows.

#### Input Interpretation

This window shows the **interpreted input** by the calculator. It shows the **two equations** for which the intersection point is required. This helps the user to confirm the input for correct results.

#### Result

This window shows the x and y coordinates of the **intersection point** of the two lines. The calculator computes the intersection point by the substitution and elimination method.

The intersection point is the point common in both the lines. It is also known as the **solution** for both the lines as both the equations satisfy the intersection point.

For the default equations y = 3x + 2 and y = 2x – 1 set by the calculator, the **intersection point** displayed in the result’s window is as follows:

\[ x = – \ 3 \]

\[ y = – \ 7 \]

The Result window also shows the option of viewing a detailed solution of the problem labeled as **Need a step-by-step solution for this problem?** By pressing on it, the user can acquire all the **mathematical steps** needed to calculate the displayed result by the calculator.

## Solved Examples

Here are some solved examples for the Intersection Calculator.

### Example 1

For the two linear equations,

**x + y = 3**

**3x – 2y = 4 **

Calculate the point of intersection between the two lines.

### Solution

The user enters the **two linear equations** in the input window one by one. The user presses “Submit” for the calculator to compute the intersection point.

The calculator displays “**intersections**” with the two equations in the input interpretation window. The equations are the same as entered by the user.

In the **Result** window, it shows the x and y coordinates for the intersection point of the two lines. The calculator uses the **elimination **and** substitution **method and computes the result as follows:

**x = 2 **

** y = 1 **

Hence, the **point of intersection** for the linear equations x + y = 3 and 3x -2y = 4 is (2,1).

### Example 2

Compute the intersecting point of the two linear equations given as:

**4x – 3y = 1 **

**x – 2y = – 6 **

### Solution

At first, the user enters the **equations** for the two lines for which the intersection point is required. To get the result, the user submits the input equations and the calculator starts computing the x and y coordinates for the point of intersection.

The **input interpretation** window shows the input equations assumed by the calculator. The user can verify the input equations from this window.

The **Result** window shows the intersection point in terms of two variables x and y. Both the equations satisfy the result given by the calculator. The (x,y) coordinates of the intersection point are the same for both equations.

The result displayed by the calculator for the above linear equations is as follows:

**x = 4 **

** y = 5 **

So the **intersection point** for the two line 4x – 3y = 1 and x – 2y = – 6 is (4,5).