1-x^12 (Linear Equations) pls give write answers
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Linear equations are equations of the first order. The linear equations are defined for lines in the coordinate system. When the equation has a homogeneous variable of degree 1 (i.e. only one variable), then it is known as a linear equation in one variable. A linear equation can have more than one variable. If the linear equation has two variables, then it is called linear equations in two variables and so on. Some of the examples of linear equations are 2x – 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, 3x – y + z = 3. In this article, we are going to discuss the definition of linear equations, standard form for linear equation in one variable, two variables, three variables and their examples with complete explanation.
Step-by-step explanation:
Linear Equation Definition
An equation is a mathematical statement, which has an equal sign (=) between the algebraic expression. Linear equations are the equations of degree 1. It is the equation for the straight line. The solutions of linear equations will generate values, which when substituted for the unknown values, make the equation true. In the case of one variable, there is only one solution. For example, the equation x + 2 = 0 has only one solution as x = -2. But in the case of the two-variable linear equation, the solutions are calculated as the Cartesian coordinates of a point of the Euclidean plane.
Below are some examples of linear equations in one variable, two variables and three variables:
Linear Equation in One variable Linear Equation in Two variables Linear Equation in Three variables
3x+5=0
(3/2)x +7 = 0
98x = 49
y+7x=3
3a+2b = 5
6x+9y-12=0
x + y + z = 0
a – 3b = c
3x + 12 y = ½ z
Forms of Linear Equation
The three forms of linear equations are
Standard Form
Slope Intercept Form
Point Slope Form
Now, let us discuss these three major forms of linear equations in detail.
Standard Form of Linear Equation
Linear equations are a combination of constants and variables. The standard form of a linear equation in one variable is represented as
ax + b = 0, where, a ≠ 0 and x is the variable.
The standard form of a linear equation in two variables is represented as
ax + by + c = 0, where, a ≠ 0, b ≠ 0 , x and y are the variables.
The standard form of a linear equation in three variables is represented as
ax + by + cz + d = 0, where a ≠ 0, b ≠ 0, c ≠ 0, x, y, z are the variables.
Slope Intercept Form
The most common form of linear equations is in slope-intercept form, which is represented as;
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.
For example, y = 3x + 7:
slope, m = 3 and intercept = 7
If a straight line is parallel to the x-axis, then the x-coordinate will be equal to zero. Therefore,
y=b
If the line is parallel to the y-axis then the y-coordinate will be zero.
mx+b = 0
x=-b/m
Slope: The slope of the line is equal to the ratio of the change in y-coordinates to the change in x-coordinates. It can be evaluated by:
m = (y2-y1)/(x2-x1)
So basically the slope shows the rise of line in the plane along with the distance covered in the x-axis. The slope of the line is also called a gradient.
Point Slope Form
In this form of linear equation, a straight line equation is formed by considering the points in the x-y plane, such that:
y – y1 = m(x – x1 )
where (x1, y1) are the coordinates of the point.
We can also express it as:
y = mx + y1 – mx1
Summary:
There are different forms to write linear equations. Some of them are:
Linear Equation General Form Example
Slope intercept form y = mx + b y + 2x = 3
Point–slope form y – y1 = m(x – x1 ) y – 3 = 6(x – 2)
General Form Ax + By + C = 0 2x + 3y – 6 = 0
Intercept form x/a + y/b = 1 x/2 + y/3 = 1
As a Function f(x) instead of y
f(x) = x + C
f(x) = x + 3
The Identity Function f(x) = x f(x) = 3x
Constant Functions f(x) = C f(x) = 6
Where m = slope of a line; (a, b) intercept of x-axis and y-axis