Important Formulas: Linear Equations | Quantitative Aptitude (Quant) - CAT PDF Download

Formulas & Definitions for Linear Equations

• A linear equation is an algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line.
• Standard form of linear equation is y = mx + b. Where, x is the variable and y, m, and b are the constants.

Forms of Linear Equations

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:

• A and B are coefficients (constants) representing the coefficients of x and y terms, respectively.
• C is a constant term.

The standard form requires that A and B are both integers and that A is non-negative. 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. Slope-Intercept Form

The slope-intercept form of a linear equation is written as:
y = mx + b
Where:

• m is the slope of the line, representing the rate of change between y and x.
• b is the y-intercept, which is the value of y when x is equal to 0. It represents the point where the line intersects the y-axis.

This form is particularly useful for graphing linear equations and quickly identifying the slope and y-intercept of the line.

3. Point-Slope Form

The point-slope form of a linear equation is given by:
y − y₁ = m(x − x₁)
Where:

• m is the slope of the line, as explained in the slope-intercept form. 
• (x₁ ,y₁) represents the coordinates of a point on the line.

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 y-intercept.

Linear equations in one variable

• A Linear Equation in one variable is defined as ax + b = 0
• Where, a and b are constant, a ≠ 0, and x is an unknown variable
• The solution of the equation ax + b = 0 is x = – b/a. We can also say that  – b/a is the root of the linear equation ax + b = 0.

Linear equations in two variable

• A Linear Equation in two variables is defined as ax + by + c = 0
• Where a, b, and c are constants and also, both a and b ≠ 0

Linear equations in three variable

• A Linear Equation in three variables is defined as ax + by + cz = d
• Where a, b, c, and d are constants and also, a, b and c ≠ 0

Formulas and Methods to solve Linear equations

Substitution Method

• Step 1: Solve one of the equations either for x or y.
• Step 2: Substitute the solution from step 1 into the other equation.
• Step 3: Now solve this equation for the second variable.

Elimination Method

• Step 1: Multiply both the equations with such numbers to make the coefficients of one of the two unknowns numerically same.
• Step 2: Subtract the second equation from the first equation.
• Step 3: In either of the two equations, substitute the value of the unknown variable. So, by solving the equation, the value of the other unknown variable is obtained.

Cross-Multiplication Method
Suppose there are two equation,

P₁x+Q₁Y = r₁ .........(1)

P₂x + Q₂Y = 12 ........(2)
Multiply Equation (1) with p₂
Multiply Equation (2) with p₁
P₁P₂x + Q₁P₂Y = r1P₂
P₁P₂X + P₁Q₂Y = P1r₂
Subtracting,
Q₁P₂Y-P₁Q₂Y = r₁P₂-P₁r₂
or, y (Q₁ P₂- Q₂P₁) = r₂P₁ - r₁P₂

Multiply Equation (1) with q₂
Multiply Equation (2) with q₁
P₁Q₂x + Q₁Q₂Y = r₁Q₂
Q₁P₂x + Q₁Q₂Y = Q₁r<sub>2
Subtracting, 

From equations (3) and (4), we get,

Note: Shortcut to solve this equation will be written as</sub>

_{Important Formulas of Linear Equation & key points to Remember}

Suppose, there are two linear equations: a₁x + b₁y = c₁ and a₂x + b₂y = c₂
Then,

Questions on Formulas of Linear Equation

Q1: What is the slope-intercept 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)
Sol: The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept.

Q2: 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)
Explanation: 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.