Revision: Linear Equations in Two Variables Algebra Maths 1 SSC (English Medium) 10th Standard Maharashtra State Board

An equation which contains two variables and the degree of each term containing a variable is one is called a linear equation in two variables.  

General Form:

ax + by + c = 0 

In simple words, all three must hold:

• It has two variables (usually x and y).
• Every variable term has power 1 (no x², xy, 1/x, √x).
• It can be written as ax + by + c = 0, with a and b not both zero.

A system of simultaneous linear equations is a pair of linear equations in the same two variables that must be satisfied together.

The standard form is:

A solution is an ordered pair (x,y) that satisfies both equations.

The graph of a linear equation in two variables is the set of all points (x,y) whose coordinates satisfy the equation.

The solution of a pair of simultaneous linear equations obtained by drawing the graph of both equations on the same coordinate plane and identifying their common point.

If the two lines intersect at a point P(x,y) then x and y satisfy both equations. Therefore, P(x,y) is the solution of the system.

A determinant is a scalar value associated with a square matrix.

For a \[ 2 \times 2 \] matrix: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

its determinant is written as: \[ |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} \]

and calculated as: \[ |A| = ad - bc \]

Remember: A determinant is a number, while a matrix is an arrangement of numbers.

A pair of equations that is not linear in the original variables xxx and yyy, but becomes a pair of linear equations after a suitable substitution, is called a pair of equations reducible to linear equations.

Cramer’s Rule

For the system: \[ a_1x + b_1y = c_1 \], \[ a_2x + b_2y = c_2 \]

\[ D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \]

\[ D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} \]

\[ D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} \]

If \[ D \neq 0, \] then the system has a unique solution:

\[ x = \frac{D_x}{D} \]

\[ y = \frac{D_y}{D} \]

• General form: ax + by + c = 0, a and b not both zero.
• Every variable term has degree 1.
• The graph is a straight line.
• To reach the general form, shift all terms to one side.

• Simultaneous equations must be solved together.

• A solution must satisfy both equations.

• Substitution replaces one variable using an expression from the other equation.

• Elimination removes one variable by adding or subtracting suitably modified equations.

• A linear equation in two variables has the form \[ ax + by + c = 0, \] where \[ a \] and \[ b \] are not both zero.
• Its graph is always a straight line.
• Every solution is an ordered pair \[ (x, y). \]
• Prepare a table of values before plotting.
• Plot at least three points to verify accuracy.
• \[ y = k \] is a horizontal line parallel to the X-axis.
• \[ x = k \] is a vertical line parallel to the Y-axis.
• A line through the origin has the form \[ ax + by = 0. \]

• A linear equation in two variables represents a straight line.

• A pair of linear equations can be solved by drawing both lines on the same graph.

• The intersection point gives the unique solution.

• A determinant is a number associated with a square matrix.
• For \[ \begin{vmatrix} a & b \\ c & d \end{vmatrix}, \] the value is \[ ad - bc. \]
• Cramer’s Rule solves a pair of linear equations using determinants.
• \[ D \] is formed from the coefficients of \[ x \] and \[ y. \]
• \[ D_x \] replaces the \[ x\text{-coefficient} \] column with constants.
• \[ D_y \] replaces the \[ y\text{-coefficient} \] column with constants.
• If \[ D \neq 0, \] the solution is \[ x = \frac{D_x}{D}, y = \frac{D_y}{D}. \]
• If \[ D = 0, \] Cramer’s Rule does not give a unique solution.

• Some equations are not linear in the given variables.

• By a suitable change of variables, they can be reduced to linear equations.

• After substitution, the equations become linear in the new variables.

• Denominators must not be zero.