Solve the simultaneous equation by substitution method [ 3 x+4 y=7 & 3 x-4 y=7 5 x-2 y=7 & 5 x-2 y=7 ]

To solve the system of equations using the substitution method, we first need to solve one of the equations for one variable in terms of the other variable. Let's solve the first equation for \(x\) in terms of \(y\):<br /><br />\[3x + 4y = 7\]<br /><br />\[3x = 7 - 4y\]<br /><br />\[x = \frac{7 - 4y}{3}\]<br /><br />Now, we substitute this expression for \(x\) into the second equation:<br /><br />\[5\left(\frac{7 - 4y}{3}\right) - 2y = 7\]<br /><br />Simplify the equation:<br /><br />\[\frac{5(7 - 4y)}{3} - 2y = 7\]<br /><br />\[\frac{35 - 20y}{3} - 2y = 7\]<br /><br />Multiply through by 3 to clear the fraction:<br /><br />\[35 - 20y - 6y = 21\]<br /><br />Combine like terms:<br /><br />\[35 - 26y = 21\]<br /><br />Subtract 35 from both sides:<br /><br />\[-26y = 21 - 35\]<br /><br />\[-26y = -14\]<br /><br />Divide by -26:<br /><br />\[y = \frac{-14}{-26}\]<br /><br />\[y = \frac{7}{13}\]<br /><br />Now that we have the value of \(y\), we can substitute it back into the expression for \(x\):<br /><br />\[x = \frac{7 - 4\left(\frac{7}{13}\right)}{3}\]<br /><br />\[x = \frac{7 - \frac{28}{13}}{3}\]<br /><br />\[x = \frac{\frac{91}{13} - \frac{28}{13}}{3}\]<br /><br />\[x = \frac{\frac{63}{13}}{3}\]<br /><br />\[x = \frac{63}{39}\]<br /><br />\[x = \frac{7}{13}\]<br /><br />So, the solution to the system of equations is \(x = \frac{7}{13}\) and \(y = \frac{7}{13}\).