Linear Equation Formula
Using linear equation formula we convert the given situation into mathematical statements illustrating the relationship between the unknowns (variables) from the information provided. Linear Equation is the equation of a straight line and is also known as equations of the first order. We use linear equations in solving our day-to-day problems. Let us understand the linear equation formula in detail in the following sections.
What is Linear Equation Formula?
An equation with the maximum order of 1 and an equation of a straight line is called the linear equation. The linear equation formula can be written in a simple slope-intercept form i.e. y = mx + b, where x and y are the variables, m is the slope of the line, and b, the y-intercept. A slope gets the direction of the line and determines how steep is the line. The value of y when x is 0 is called the y-intercept because (0,y) is the point at which the line crosses the y-axis. Hence, the linear equation formula is given by:
y = mx + b
where,
- x and y are two variables
- b is the y-intercept
- m is the slope of the line
Linear Equation Formula
Applying the linear equation formula(y= mx+b), the equation of a straight line that has y-intercept (0, 2) with the slope m= 4 is given by y = 4x+2. The linear equation in one variable, and in two variables can be represented in many forms where a line can be defined in an (x,y) plane. Some of the common forms are:
| Linear Equation Form | Formula | Explanation |
| General Form |
ax + by = c |
where a ≠ 0; a, b, c are real numbers. m = -a/b |
| Slope Intercept Form | y = mx + b | where y and x are the points in the (x,y) plane, m is the slope of the line (knows as the gradient), b is the intercept (a constant value) |
|
\(y-y_{1} = m (x-x_{1})\) |
where \((x_{1},y_{1})\) and \((x_{2}, y_{2})\) are the two points on the line, m= \(\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\) and \(x_{1}\) ≠ \(x_{2}\) |
|
| Intercept Form | x/\(x_{0}\) + y/\(y_{0}\)= 1 |
where \(x_{0}\) = x-intercept \(y_{0}\) = y-intercept |
| Vertical Line | x = p | p is the x-intercept |
| Horizontal Line | y = q | q is the y-intercept |
The linear equations can be solved for a specific variable by the substitution method or elimination method or graphically. Let us look at the linear equations in the following graph as seen below:
Derivation of Linear Equation Formula
Let us see the linear equation formula determined on a graph in a straight line or slope. A slope is a line equal to the ratio of change in y-coordinates and change in x-coordinates.
The equation of a straight line is given by: y = mx + b
Where m is the slope of the line, b is the y-intercept, x and y are the coordinates of the x-axis and y-axis, respectively.
When the straight line is parallel to the x-axis, then the x-coordinate will be equal to zero. Therefore, y = b
When the straight line is parallel to the y-axis then the y-coordinate will be zero. Therefore,
mx + b = 0
x = -b/m
So basically the slope shows the rise of line in the plane along with the distance covered in the x-axis.
Examples Using Linear Equation Formula
Example 1: Jake's piggy bank has11 coins (only quarters or dimes) that have a value of S1.85. How many dimes and quarters does the piggy bank has?
Solution: Let us assume that:
Number of dimes = x
Number of quarters = y
Since there are 11 coins in total, x + y = 11 ⇒ y = 11 - x → (1)
The total value of the money in the piggy bank is $1.85.
We know that 100 cent, = $1, 1 quarter = $1 and 10 dimes = $1
Thus we get 10x + 25y = 185 → (2)
Let us follow the substitution method and solve the equations (1) and (2)
10x + 25(11 − x) = 185 (Substituting (1) in (2))
10x + 275 − 25x = 185
−15x + 275 = 185
−15x = −90
x = 6
Substitute this in (1): y = 11 - 6 = 5
Therefore, the number of dimes = 6; The number of quarters = 5
Example 2: John gets the equation of a straight line 3x + 4y = 15. Help him find the slope.
Solution: Given 3x + 4y = 15
This equation is of the general form: ax+by = c
in the equation 3x + 4y = 15, a = 3 and b = 4
Using the linear equation formula, we get the slope as m = -a/b
m = -3/4
Thus the slope of this straight line is m = -3/4
Example 3: Find the value of y from the equations, 2x + 4y = 20 when x = 24
Solution:
Put the value of x in the equation and solve for y.
Given, x = 24 and 2x + 4y = 20
4y = 20 - 2 × 24
4y = -28
y = -7
Therefore, the value of y = -7
FAQs on Linear Equation Formulas
What is Linear Equation Formula?
The linear equation formula is obtained based on in which form is the equation of the straight line is.
What is the Formula for Calculating a Linear Equation?
The formula to calculate the linear equation on a slope is:
y = mx + b
where,
- x and y are two variables
- b is the y intercept
- m is the slope of the line
What are the Different Forms Used in the Linear Equation Formula?
The different forms used in the linear equation formula where the line can be plotted on a graph on both x-axis and y-axis, are:
- General Form
- Slope Intercept Form
- Point Form
- Intercept Form
- Two-Point form
What is the Standard Form of Linear Equation Formula?
The standard form of the linear equation formula consists of both the variable and a constant. The formula is ax + b = 0 where a ≠ 0 and x is the variable.
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