Solution of a Linear Equation
This section discusses the solutions to linear equations in two variables. Unlike linear equations in one variable that possess a unique solution, a linear equation in two variables provides us not just with one solution, but rather infinitely many pairs of values satisfying the equation form.
For example, the equation
$$2x + 3y = 12$$ is examined, demonstrating how different pairs of $(x, y)$ sets, such as (3, 2) and (0, 4), fulfill the equation's requirements. Students learn to verify candidates' solutions through substitution and are encouraged to explore choosing values for one variable to find corresponding values for the other variable.
The section includes practical examples to derive multiple solutions from a single linear equation, explaining methods like setting one variable to zero to find intercepts on the Cartesian plane. Through engaging exercises and examples, learners understand that each valid solution corresponds to a point on the linear equation's graph, revealing the infinite nature of solutions in this context.
Example : Find four different solutions of the equation \(x + 3y = 12\).
Solution: By inspection, \(x = 6, y = 2\) is a solution because for \(x = 6, y = 2\)
\[ 6 + 3 \times 2 = 12 \]
Now, let us choose \(x = 0\). With this value of \(x\), the equation reduces to \(3y = 12\) which has the unique solution \(y = 4\) so, \(x = 0, y = 4\) is also a solution of the given equation.
Similarly, taking \(y = 0\), the given equation reduces to \(x = 12\). So, \(x = 12, y = 0\) is a solution of \(x + 3y = 12\). Finally, let us take \(y = 1\). The given equation now reduces to \(x + 3 = 12\), whose solution is \(x = 9\). Therefore, \((9, 1), (6, 2), (0, 4), (12, 0)\) is also a solution of the given equation. So four of the infinitely many solutions of the given equation are:
\[(6, 2), (0, 4), (12, 0), (9, 1)\]