Pair of Linear Equations in two variables
Linear equations in two variables are equation which can be expressed as ax + by + c
= 0, where a, b and c are real numbers and both a, and b are not zero. The solution of
such equations is a pair of values for x and y which makes both sides of the equation
equal.
Let’s look at the solutions of some linear equations in two variables. Consider the
equation 2x + 3y = 5. There are two variables in this equation, x and y.
Scenario 1: Let’s substitute x = 1 and y = 1 in the Left Hand Side (LHS) of the
equation. Hence, 2(1) + 3(1) = 2 + 3 = 5 = RHS (Right Hand Side). Hence, we
can conclude that x = 1 and y = 1 is a solution of the equation 2x + 3y = 5.
Therefore, x = 1 and y = 1 is a solution of the equation 2x + 3y = 5.
Scenario 2: Let’s substitute x = 1 and y = 7 in the LHS of the equation. Hence,
2(1) + 3(7) = 2 + 21 = 23 ≠ RHS. Therefore, x = 1 and y = 7 is not a solution of
the equation 2x + 3y = 5.
, Geometrically, this means that the point (1, 1) lies on the line representing the
equation 2x + 3y = 5. Also, the point (1, 7) does not lie on this line. In simple words,
every solution of the equation is a point on the line representing it.
To generalize, each solution (x, y) of a linear equation in two variables, ax + by + c =
0, corresponds to a point on the line representing the equation, and vice versa.
Forms of Linear Equation
A line is determined in an X-Y plane through various forms. Here, we have a list
of some common conditions which we use to solve linear equations.
Slope Intercept Form
General Form
Intercept Form
Two-point Form
Point Form
The Standard Form of Linear Equations
The combination of variables and constants form a linear equation. A linear
equation with one variable is depicted as ax+b=0, where x is a variable, and a≠0.
The standard form of linear equation with two variables is depicted as ax+by+c=0,
where a is not equal to zero, b is not equal to zero, and x and y are variables.
The standard form of a linear equation with three variables is depicted as
ax+by+cz+d=0, where a, b and c are not equal to zero and x, y, z are variables.
Slope Intercept Form
The most common way to solve linear equations is in the slope-intercept form. It is
represented as y=mx+c, where y and x are the x-y plane points, c is the intercept
with a constant value and m is the slope of the line known as a gradient.
For example, y=5x+9
Slope, m=5, and intercept=9.
Point Slope Form
Linear equations in two variables are equation which can be expressed as ax + by + c
= 0, where a, b and c are real numbers and both a, and b are not zero. The solution of
such equations is a pair of values for x and y which makes both sides of the equation
equal.
Let’s look at the solutions of some linear equations in two variables. Consider the
equation 2x + 3y = 5. There are two variables in this equation, x and y.
Scenario 1: Let’s substitute x = 1 and y = 1 in the Left Hand Side (LHS) of the
equation. Hence, 2(1) + 3(1) = 2 + 3 = 5 = RHS (Right Hand Side). Hence, we
can conclude that x = 1 and y = 1 is a solution of the equation 2x + 3y = 5.
Therefore, x = 1 and y = 1 is a solution of the equation 2x + 3y = 5.
Scenario 2: Let’s substitute x = 1 and y = 7 in the LHS of the equation. Hence,
2(1) + 3(7) = 2 + 21 = 23 ≠ RHS. Therefore, x = 1 and y = 7 is not a solution of
the equation 2x + 3y = 5.
, Geometrically, this means that the point (1, 1) lies on the line representing the
equation 2x + 3y = 5. Also, the point (1, 7) does not lie on this line. In simple words,
every solution of the equation is a point on the line representing it.
To generalize, each solution (x, y) of a linear equation in two variables, ax + by + c =
0, corresponds to a point on the line representing the equation, and vice versa.
Forms of Linear Equation
A line is determined in an X-Y plane through various forms. Here, we have a list
of some common conditions which we use to solve linear equations.
Slope Intercept Form
General Form
Intercept Form
Two-point Form
Point Form
The Standard Form of Linear Equations
The combination of variables and constants form a linear equation. A linear
equation with one variable is depicted as ax+b=0, where x is a variable, and a≠0.
The standard form of linear equation with two variables is depicted as ax+by+c=0,
where a is not equal to zero, b is not equal to zero, and x and y are variables.
The standard form of a linear equation with three variables is depicted as
ax+by+cz+d=0, where a, b and c are not equal to zero and x, y, z are variables.
Slope Intercept Form
The most common way to solve linear equations is in the slope-intercept form. It is
represented as y=mx+c, where y and x are the x-y plane points, c is the intercept
with a constant value and m is the slope of the line known as a gradient.
For example, y=5x+9
Slope, m=5, and intercept=9.
Point Slope Form
