A linear equation, synonymous with an algebraic equation where each term carries an exponent of one, is graphically represented by a straight line. The standard form of a linear equation is y = mx + b, where x is the variable, and y, m, and b are constants.
There are mainly 3 forms of Linear Equation :
1. Standard Form
The standard form of a linear equation is typically written as:
Ax + By = C
Where:
The standard form requires that A and B are both integers and that A is nonnegative. Also, A and B should not have any common factors other than 1. This form is commonly used in algebraic manipulation and solving systems of linear equations.
2. SlopeIntercept Form
The slopeintercept form of a linear equation is written as:
y=mx+b
Where:
This form is particularly useful for graphing linear equations and quickly identifying the slope and yintercept of the line.
3. PointSlope Form
The pointslope form of a linear equation is given by:
y − y_{1} = m(x − x _{1})
Where:
This form is useful when you know a specific point on the line and its slope, allowing you to write the equation directly without having to calculate the yintercept.
Substitution Method
Elimination Method
CrossMultiplication Method
Suppose there are two equation,
Multiply Equation (1) with p_{2}
Multiply Equation (2) with p_{1}
Subtracting,
where (p_{1}q_{2} – p_{2}q_{1}) ≠ 0
Multiply Equation (1) with q_{2}
Multiply Equation (2) with q_{1}
Subtracting,
where (p_{1}q_{2} – p_{2}q_{1}) ≠ 0
From equations (3) and (4), we get,
where (p_{1}q_{2} – p_{2}q_{1}) ≠ 0
Note: Shortcut to solve this equation will be written as
which means,
Suppose, there are two linear equations: a_{1}x + b_{1}y = c_{1} and a_{2}x + b_{2}y = c_{2}
Then,
((a) If then there will be one solution, and the graphs will have intersecting lines.
((b) If then there will be numerous solutions, and the graphs will have coincident lines.
((c) If then there will be no solution, and the graphs will have parallel lines.
Q1: What is the slopeintercept form of the equation of a line?
(a) y=mx+b
(b) y=mx−b
(c) y=bx+m
(d) y=bx−m
Ans: (a)
The slopeintercept form of a linear equation is y=mx+b, where m represents the slope and b represents the yintercept.
Q2: What does the standard form of a linear equation look like?
(a) y = mx+b
(b) y = bx+m
(c) Ax + By=C
(d) Ax + By=D
Ans: (c)
The standard form of a linear equation is Ax+By=C, where A and B are coefficients representing the coefficients of x and y, respectively, and C is a constant term.
Q3: How can you eliminate fractions from a linear equation?
(a) Multiply both sides of the equation by a common denominator
(b) Divide both sides of the equation by a common denominator
(c) Add both sides of the equation by a common denominator
(d) Subtract both sides of the equation by a common denominator
Ans: (a)
To eliminate fractions from a linear equation, multiply both sides of the equation by a common denominator. This process will clear the fractions and make the equation easier to solve.
Q4: Which form of a linear equation is useful when you know a specific point on the line and its slope?
(a) PointSlope Form
(b) SlopeIntercept Form
(c) Standard Form
(d) None of the above
Ans: (a)
The pointslope form of a linear equation is written as y − y_{1} =m(x − x_{1}), where (1,1)(x_{1}, y_{1}) represents the coordinates of a point on the line, and m is the slope.
Q5: Which of the following is the correct representation of the pointslope form of a linear equation?
(a) y = mx + b
(b) y − y_{1 }= m(x−x_{1})
(c) y = Ax + By
(d) y = m_{1}x + b
Ans: (b)
The pointslope form of a linear equation is represented as y − y_{1 }= m(x−x_{1}) , where (x_{1} , y_{1}) represents the coordinates of a point on the line, and m is the slope of the line. This form is useful when you know a specific point on the line and its slope.
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