Solving Linear Equations
2026-02-28 08:55 Diff
https://brightchamps.com/en-us/math/algebra/solving-linear-equations

There are many methods for solving linear equations. Some of the methods are:
 

  • Graphical Method
  • Elimination Method
  • Substitution Method
  • Cross Multiplication Method
  • Matrix Method
  • Determinants Method (Cramer’s Rule) 

Graphical Method

For solving linear equations in a graphical method, we draw the equations as a line on a graph. The point where the two lines cross or intersect is the answer.

Example:
\(y = 2x + 1\)
\(y = x + 3\)


Plot both lines on the graph.
Where they meet is the answer.
If they meet at (2, 5), that means x = 2 and y = 5 is the solution.

Elimination Method

In this method, we add or subtract two equations to cancel out one variable, allowing us to find the value of the remaining variable.
 

Example:
\(x + y = 10\)
\(x - y = 4\)
Add both equations,
\((x + y) + (x - y) = 10 + 4\)
\(2x = 14\)
\(x = 7\)
Now put x = 7 in the first equation:
\(7 + y = 10\)
\(y = 3\)
So the final answer is \(x = 7, y = 3\)

Substitution Method

Substitution method means solving one equation to find one variable and then substituting it into the other.


Example:
Solve the system:
(1) \(y = x + 2\)
(2) \(x + y = 10\)


Step 1: Use equation (1) to substitute for y in equation (2):
\(x + (x + 2) = 10\)


Step 2: simplify the equation
\(2x + 2 = 10\)
\(2x = 8\)
\(x = 4\)


Step 3: Substitute \(x = 4\) back into equation (1):
\(y = 4 + 2 = 6\)
Answer: \(x = 4, y = 6\)

Cross Multiplication Method

For solving linear equations using the cross multiplication method, we use formulas. The formula for solving two equations in the form of:

\(a_1x + b_1y = c_1\)

\(a_2x + b_2y = c_2\)

\(x = \frac{(b1c2 - b2c1)}{(a1b2 - a2b1)}, y = \frac{(c1a2 - c2a1)}{(a1b2 - a2b1)}\)

Matrix Method

A matrix method is a neat way to write equations in rows and columns. Let's see this using an example.

\(x + y = 6\)
\(2x + 3y = 14\)


Write this in a matrix form as:
 

\( A = \begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} 6 \\ 14 \end{bmatrix} \)

Now use the formula:
\(AX = B\)
\(X = A^{-1}B\)

Once we substitute the values and solve the equation, we get \(x = 4, y = 2\).

Determinants Method (Cramer’s Rule)

This method uses determinants, which are specific numerical values calculated from square matrices and help in solving systems of equations.


Example:
\(2x + 3y = 12\)
\(x - y = 1\)

We find three determinants:
\( \Delta \) from coefficients
\(\Delta_1\) replace the first column with answer numbers
\(\Delta_2\) replace second column with answer numbers

\(x = \frac{\Delta_1}{\Delta}, y =\frac{\Delta_2}{\Delta}\)