Pair of Linear Equations in Two Variables – Formula & Equation Sheet
Essential formulas and equations from Mathematic, tailored for Class 10 in Mathematics.
This one-pager compiles key formulas and equations from the Pair of Linear Equations in Two Variables chapter of Mathematic. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
y = mx + c
y is the dependent variable, m is the slope (change in y/change in x), x is the independent variable, and c is the y-intercept (value of y when x=0). This equation represents a line in the Cartesian plane.
Slope (m) = (y₂ - y₁) / (x₂ - x₁)
This formula calculates the slope between two points (x₁, y₁) and (x₂, y₂). A steeper slope indicates a steeper line on the graph.
x₁/a₁ = x₂/a₂ = y₁/b₁ = y₂/b₂
This expresses the condition under which two lines are either parallel or coincident. Here, (x₁, y₁) and (x₂, y₂) are coordinates of points on lines with slopes a₁ and a₂ respectively.
y - y₁ = m(x - x₁)
This point-slope form of the equation of a line allows you to write the equation of a line given a point (x₁, y₁) and the slope m.
General form: ax + by + c = 0
This is the standard form of a linear equation in two variables, where a, b, and c are real numbers. This form is often used for solving systems of equations.
x + y = k
This represents a line where the sum of x and y is constant (k). Useful for easily finding intercepts.
Elimination Method: a₁x + b₁y = c₁ and a₂x + b₂y = c₂
Utilizes addition or subtraction to eliminate one variable and solve for the other, making it efficient for finding solutions of linear equations.
Substitution Method: Solve for x or y in one equation, and substitute into the other.
This method replaces one variable with an expression from another equation, simplifying the system for easier solving.
Graphical Method: Plot the equations on a graph.
Visualize the solutions by plotting both equations. The intersection point(s) represent the solution(s) of the equations.
Infinite solutions criterion: If a₁/a₂ = b₁/b₂ = c₁/c₂
Determines that the lines are coincident (same line), thus having infinitely many solutions.
Equations
3x + 4y = 20
This equation represents a linear relationship between the variables x and y. It can be solved using various methods for specific solutions.
2x + 3y = 9
A linear equation representing another line in the same two-dimensional space. Finding solutions involves intersection with another line.
x – 2y = 0
This equation can help derive the relationship between x and y where y is directly proportional to x, ideal for linear relationships.
y = (1/2)x
This equation indicates that y is half of x, easily demonstrates proportionality and linearity.
x + 2y = 8
This equation describes a line where the sum of x and twice y equals 8, useful in graphical interpretations.
5x – 3y = 12
A linear equation that can be used to find specific values of x and y through methodical solving.
2x - 5 = y
This rearranged form shows y in terms of x, allowing for direct calculation of y values from given x.
x + y = 6
Indicates a line where the sum of x and y is constant, important in algebraic applications.
4x + y = 24
Another standard linear equation form to find relationships between x and y across a range of values.
y = 3x + 2
This slope-intercept form indicates that the line crosses the y-axis at 2 with a slope of 3, illustrating direction.
