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 

A determinant is a number associated with a square matrix.

\[\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}=ad-bc\]

The value of the determinant is ad - bc.

The degree of a 2 × 2 determinant is 2.

Cramer’s Rule is a method to solve simultaneous linear equations using determinants.

• It can be applied only when the determinant D ≠ 0

• $a1x+b1y=c1$

\[D=
\begin{vmatrix}
a_1 & b_1 \\
a_2 & b_2
\end{vmatrix}=a_1b_2-a_2b_1\]

\[D_x=
\begin{vmatrix}
c_1 & b_1 \\
c_2 & b_2
\end{vmatrix}=c_1b_2-c_2b_1\]

\[D_y=
\begin{vmatrix}
a_1 & c_1 \\
a_2 & c_2
\end{vmatrix}=a_1c_2-a_2c_1\]

\[x=\frac{D_x}{D}\quad\mathrm{and}\quad y=\frac{D_y}{D}\]

• If D ≠ 0 → unique solution

• If D = 0 → Cramer’s rule is not applicable

Simultaneous Linear Equations: Two linear equations solved together.

Elimination method:

1. Make the coefficients equal
2. Add/subtract 
3. Find one variable
4. Substitute to get the other.

Substitution method:

1. Express one variable in terms of the other, 
2. Substitute 
3. Solve 
4. Substitute back.

• The graph of a linear equation is a straight line.

• Find at least 4 ordered pairs for the given equation.

• Draw the X-axis and Y-axis with a suitable scale.

• Plot all ordered pairs on the graph.

• Check that all points lie on one straight line.

• Join the points to obtain the graph.

• Equation y = k → line parallel to the X-axis.

• Equation x = k → line parallel to the Y-axis.

• Each linear equation represents a straight line.

• Find ordered pairs and plot both lines on the same graph.

• The point of intersection gives the solution.

• Intersecting lines → one solution

• Parallel lines → no solution

• Coincident lines → infinitely many solutions

• 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.