** (9, 3), (-6, 7y), m = 3**

This **question aims** to find unknown points from **two points and slopes**. A **two-point form** can **express the equation of a straight line** in a **coordinate plane**. **Equation of a line can be found by various methods depending on the information available.** The **two-point form is one of the methods**. This is used to find the equation of a line when two points lying on the line are given. Some other important forms to represent the equation of a line are **slope-intercept form**, **intercept form**, **point-slope form**, etc.

The two-point form is one of the important forms used to represent a straight line algebraically. The** equation of a line represents** every point on the line, i.e., it is satisfied by every point on the line. The** two-point line form** is used to find the equation of a line given two points $(x1, y1)$ and $(x2,y2)$.

**Equation of a line in the form of two-points form:**

Two-point form of a line passing through these two points is given by:

\[y-y_{1}=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})\]

Where $(x,y)$ are variables and $(x_{1},y_{1}) \:and (x_{2},y_{2})$ are points on the line.

A **line passing through two points will have an equation of the form**. The **equation using two points** can also be written as:

\[y=mx+c\]

We can find the** slope value** $m$, the gradient of the line, by** creating a right triangle using the coordinates of the two given points .** We can then find the **value** of $c$, the point of intersection $y$, by substituting the coordinates of one point into the equation. The **final output can be checked by substituting the coordinates of the second point into the equation.**

**Expert Answer**

**Formula for the slope of the line, given two points on that line is given by:**

\[m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

**Plug the values of the points on the line** and the value of the **slope** to find the value of **unknown** $y$.

\[3=\dfrac{7y-3}{-6-9}\]

\[3=\dfrac{7y-3}{-15}\]

**Cross multiplying** and **solving for unknown.**

\[-45=7y-3\]

\[7y=-42\]

\[y=-6\]

The** value of the unknown** $y$ is $-6$.

**Numerical Result**

The value of the unknown $y$ for the two points and the slope is $-6$.

**Example**

**Determine the value of x or y so that the line passing through the given points has the given slope.**

** (6, 2), (-6, 2y), m = 5**

**Solution**

**Formula for the slope of the line**, **given two points on that line is given** by:

\[m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

**Plug the values of the points** on the line and the value of the** slope** to find the value of** unknown** $y$.

\[5=\dfrac{2y-2}{-6-6}\]

\[5=\dfrac{2y-2}{-12}\]

**Cross multiplying** and solving for unknown.

\[-60=2y-2\]

\[2y=-58\]

\[y=-29\]

The **value of the unknown** $y$ is $-29$.

The** value of the unknown** $y$ for the two points and the slope is $-29$.