Question

Solve the following pair of linear equation by the substitution method
\[3x - y = 3\]
\[9x - 3y = 9\]

Answer 1

Hint: Here we have a system of two linear equations with two variables. We need to find the value of ‘x’ and ‘y’. First, we need to solve one equation for one of the variables and then we need to substitute this expression into another equation and we solve it. Using this we will have one variable value and to find the other we substitute the obtained variable value in any one of the given equations.

Complete step-by-step solution:
Given,
\[3x - y = 3{\text{ }} - - - (1)\]
\[9x - 3y = 9{\text{ }} - - - (2)\]
From equation (1) we have,
\[y = 3x - 3\]
Now we substitute this ‘y’ value in equation (2) we have,
\[9x - 3(3x - 3) = 9\]
Thus, we have a linear equation with one variable and we can simplify for ‘x’,
\[9x - 9x + 9 = 9\]
\[\Rightarrow 9 = 9\]
That is, we can clearly see that the system of two equations has an infinite number of solutions, as for every value of x and y the equation $9=9$ satisfies so we have infinite solutions for the given pair of linear equations. That is the given system of equations is dependent.

Note: If we draw the graph for the given two equations, both are in the same lines. Both equations have the same y-intercept and slopes. Also, If the given system of the equation has at least one solution then it is said to be consistent. If a consistent system has exactly one solution, then it is said to be independent. We also have a system of the equation which is inconsistent and the graph of the line does not intersect, so the lines are parallel and there is no solution for the given system.