Write an equation of the line that passes through calculator

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Solution:

The general equation of a straight line can be written as y = mx + c where m is the slope and c is the y-intercept.

Let us consider the given points (0, 1) and (-2, -5).

As we know that the equation of a line passing through the points (x1, y1) and (x2, y2) is given by y - y1 = m (x - x1).

Here, m is the slope given by the formula m = (y2 - y1) / (x2 - x1). Check out Cuemath's Slope Calculator that helps you to calculate the slope.

Hence, on substituting the given points in the equation of a line, we get

y - 1 = m (x - 0)

m =  (y2 - y1) / (x2 - x1)

m = (-5 - 1) / (-2 - 0)

m = -6 / -2 = 3

Substituting the value of m in y - 1 = m (x - 0), we get

y - 1 = 3 (x - 0)

y - 1 = 3x

y = 3x + 1

y - 3x -1 = 0

You can use Cuemath's online Equation of Line calculator to find the equation of a line.

Therefore, the equation of a line passing through the points (0, 1) and (-2, -5) is y - 3x -1 = 0

Write an equation of the line that passes through the points (0, 1) and (-2, -5).

Summary:

The equation of a line through the points (0, 1) and (-2, -5) is y - 3x - 1 = 0

Solution:

A two-point form of the equation is used when two different points on the line are known.

This equation can easily be simplified to any of the forms of the equation like the slope-intercept form, so as to calculate the intercept value by comparison.

Let the given points are (x1, y1) = (1, 0) and (x2, y2) = (3, 4).

Therefore, applying the slope-intercept form of the equation,

We get,

⇒ y - y1 = m (x - x1)

⇒ m = slope formula = (y2 - y1) / (x2 - x1) 

Slope of the line = m = (4 - 0) / (3 - 1) = 4 / 2 = 2 

You can find the slope using the slope calculator.

Using the point (1, 0), let's write the equation of the line.

(y - 0) = m (x - 1) [Since, (y2 - y1) / (x2 - x1) = m]

⇒ y = 2(x - 1) 

⇒ y = 2x - 2

Thus, the equation of the line passing through the points (1, 0) and (3, 4) is y = 2x - 2.

Write an equation of the line that passes through the given two points: (1, 0) and (3, 4) 

Summary:

The general equation of the line that passes through the given two points: (1, 0) and (3, 4) is y = 2x - 2

Determine the equation of a line passing through the points $(-2, 5)$ and $(4, -2)$.

Find the slope - intercept form of a straight line passing through the points $\left( \frac{7}{2}, 4 \right)$ and $\left(\frac{1}{2}, 1 \right)$.

If points $\left( 3, -5 \right)$ and $\left(-5, -1\right)$ are lying on a straight line, determine the slope-intercept form of the line.

How to find equation of the line determined by two points?

To find equation of the line passing through points $A(x_A, y_A)$ and $B(x_B, y_B)$ ( $ x_A \ne x_B $ ), we use formula:

$$ {\color{blue}{ y - y_A = \frac{y_B - y_A}{x_B-x_A}(x-x_A) }} $$

Example:

Find the equation of the line determined by $A(-2, 4)$ and $B(3, -2)$.

Solution:

In this example we have: $ x_A = -2,~~ y_A = 4,~~ x_B = 3,~~ y_B = -2$. So we have:

$$ \begin{aligned} y - y_A & = \frac{y_B - y_A}{x_B-x_A}(x-x_A) \\ y - 4 & = \frac{-2 - 4}{3 - (-2)}(x - (-2)) \\ y - 4 & = \frac{-6}{5}(x + 2) \end{aligned} $$

Multiply both sides with $5$ to get rid of the fractions.

$$ \begin{aligned} (y - 4)\cdot {\color{red}{ 5 }} & = \frac{-6}{5}\cdot {\color{red}{ 5 }}(x + 2)\\ 5y - 20 & = -6(x + 2)\\ 5y - 20 & = -6x - 12 \\ 5y & = -6x - 12 + 20 \\ 5y & = -6x + 8 \\ {\color{blue}{ y }} & {\color{blue}{ = -\frac{6}{5}x - \frac{8}{5} }} \end{aligned} $$

In special case (when $x_A = x_B$ the equation of the line is:

$$ {\color{blue}{ x = x_A }} $$

Example 2:

Find the equation of the line determined by $A(2, 4)$ and $B(2, -1)$.

Solution:

In this example we have: $ x_A = 2,~~ y_A = 4,$ $ x_B = 2,~~ y_B = -1$. Since $x_A = x_B$, the equation of the line is:

$$ {\color{blue}{ x = 2 }} $$

You can see from picture on the right that in special case the line is parallel to y - axis.

Note: use above calculator to check the results.

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