In mathematics and physics, a linear equation is frequently referred to as an equation of the line when it is used to represent a line in cartesian coordinates. It has a significant impact on engineering as well.

In mathematics and physics, a linear
equation is frequently referred to as an equation of the line when it is used
to represent a line in cartesian coordinates. It has a significant impact on
engineering as well.

It enables engineers to determine the
anticipated load on the bridge (which is being built) at any time. In the
following sections of this article, with the aid of numerous examples, we will
examine the fundamental definition, formula, and various situations for
determining the slope-intercept form of a linear equation.

The slope-intercept form is a
technique for figuring out how to solve a linear equation. With this approach,
the slope and y-intercept are used to calculate the equation of a line. If we
don't already know these numbers, we'll figure them out first.

The slope-intercept form is typically
represented using the following equation.

**Y = mx + c **

- “x”
and “y” are the representative points, these are variables
- “c” is
the y-intercept
- “m” is
the slope of the line.

This shape can be acquired in a
variety of situations and ways.

**Typical Cases**

We can determine the slope-intercept
form in the three scenarios listed below:

1.
Using two points (two ordered pairs).

2.
Using x, y, and the slope “m”.

3.
Using the slope “m” and “c”.

**part example: **Using mathematical puzzles as a guide, we'll
explain all three examples in this part.

**Using two points
in Case 1.**

**Figure 1:**

If a line has the points (7, 4) and
(9, -6), determine the linear equation of the line. Make a slope-intercept
diagram of it.

**Solution:**

**Step 1:** Extract the given data:

x_{1} =
7, y_{1} = 4, x_{2} = 9, y_{2} = -6

**Step 2:** Compute the slope “m”.

Slope = m = Rise /
run

Slope = m = (y_{2} –
y_{1}) / (x_{2} – x_{1})

Slope = m = (-6 –
(4)) / (9 – (7))

Slope = m = (-6 – 4)
/ (9 – 7)

Slope = m = -10/2

**Slope = m = -5**

**Step 3:** Find out “c” using 1^{st} point (ordered
pair).

y = mx + c

4 = -5 (7) + c

4 = -35 + c

4 + 35 = c

**c = 39**

**Step 4:** Place all values in the general formula of the
slope-intercept form.

y = mx + c

**y = -5x + 39**

**Example 2:**

Calculate using the
slope-intercept form formula, x_{1} = 12, x_{2} = 9,
y_{1} = 2, y_{2} = 14

**Solution:**

**Step 1:** Extract the given data:

x_{1} =
12, x_{2} = 9, y_{1} = 2, y_{2} = 14

**Step 2:** Compute the slope “m”.

Slope = m = Rise /
run

Slope = m = (y_{2} –
y_{1}) / (x_{2} – x_{1})

Slope = m = (14 – 2)
/ (9 – 12)

Slope = m = (12) /
(-3)

**Slope = m = -4**

**Step 3:** Find out “c” using 1^{st} point (ordered
pair).

y = mx + c

2 = -4 (12) + c

2 = -48 + c

2 + 48 = c

**c = 50**

**Step 4:** Place all values in the general formula of the
slope-intercept form.

y = mx + c

m = -4, c = 50

**y = -4x + 50**

Case 2: Using x, y, and the slope “m”.

**Example 1:**

Evaluate y
= mx + c if m = 4, x = 2, and y = – 3.

**Solution:**

**Step 1:** Extract the given data.

y = -3, x =
2, and m = 4

**Step 2:** Calculate “c” using the given data

y = -3, x =
2, and m = 4

-3 = 4 (2)
+ c

-3 = 8 + c

-11 = c

c = -11

**Step 3:** Represent it in “y = mx + c” form

m = 4, c =
-11

y = mx + c

**y = 4x – 11**

**Example 2:**

Evaluate
the slope-intercept form of the line if it has m = 3, x = 9, and y = – 12.

Solution:

**Step 1:** Extract the given data.

y = -12, x
= 9, and m = 3

**Step 2:** Calculate “c” using the given data

y = -12, x
= 9, and m = 3

-12 = 3 (9)
+ c

-12 = 27 +
c

-39 = c

c = -39

**Step 3:** Represent it in “y = mx + c” form

m = 3, c =
-39

y = mx + c

**y = 3x – 39**

Case 3: Using the slope “m” and “c”

**Example 1:**

If c
(y-intercept) of a line is 9, and its slope m = -1 then calculate its
slope-intercept form.

**Solution:**

**Step 1: **Extract the given data.

c = 9, m =
-1

**Step 2: **Put the values in the general formula.

y = mx + c

m = -1, c =
9

**y = -x + 9**

**Summary:**

The fundamental definition, functions,
and applications of the slope-intercept form have been covered in this article.
Additionally, we have reviewed the general formula and the scenarios by which
we can assess the y = mx + c form of an equation.

You may now resolve all the issues
pertaining to this subject since we covered how to determine the
slope-intercept form in the example section.