Ex 1. ) 3x-3y=9 2x-2y=6

To solve the system of linear equations:<br /><br />\[<br />\left\{<br />\begin{array}{l}<br />3x - 3y = 9 \\<br />2x - 2y = 6<br />\end{array}<br />\right.<br />\]<br /><br />we can use either the substitution method or the elimination method. Here, we will use the elimination method.<br /><br />First, let's simplify both equations by dividing each term by the common factor in each equation:<br /><br />1. \(3x - 3y = 9\)<br /><br /> Divide each term by 3:<br /> \[<br /> x - y = 3 \quad \text{(Equation 1)}<br /> \]<br /><br />2. \(2x - 2y = 6\)<br /><br /> Divide each term by 2:<br /> \[<br /> x - y = 3 \quad \text{(Equation 2)}<br /> \]<br /><br />We observe that both simplified equations are identical:<br /><br />\[<br />x - y = 3<br />\]<br /><br />This means that the two original equations are actually the same line, and thus there are infinitely many solutions along this line. Any point \((x, y)\) that satisfies \(x - y = 3\) is a solution to the system.<br /><br />To express the solution set, we can write:<br /><br />\[<br />x = y + 3<br />\]<br /><br />So, the general solution is:<br /><br />\[<br />(x, y) = (y + 3, y)<br />\]<br /><br />where \(y\) can be any real number.