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 set of points in a plane is called a convex set if the line segment joining any two points in the set lies entirely within the set.

If the line segment joining any two points in the set does not completely lie in the set, then it is a non-convex set.

A linear inequality or inequation, which has only one variable, is called a linear inequality or inequation in one variable.

e.g. ax + b < 0, where a ≠ 0, 3x + 4 > 0

An inequality or inequation is said to be linear if each variable occurs in the first degree only and there is no term involving the product of the variables.

e.g. ax + b ≤ 0, ax + by + c > 0, x ≤ 4

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.