7. x=3y-1 and 2x-6y=-2 3. ) 4x-7y=7 12x-21y=24

Let's solve each system of equations one by one.<br /><br />### Problem 7:<br /><br />We have the following system of equations:<br />1. \( x = 3y - 1 \)<br />2. \( 2x - 6y = -2 \)<br /><br />**Step 1: Substitute Equation 1 into Equation 2**<br /><br />Since \( x = 3y - 1 \), substitute this expression for \( x \) in Equation 2:<br /><br />\[ 2(3y - 1) - 6y = -2 \]<br /><br />**Step 2: Simplify and Solve for \( y \)**<br /><br />\[ 6y - 2 - 6y = -2 \]<br /><br />This simplifies to:<br /><br />\[ -2 = -2 \]<br /><br />This equation is always true, which means that the two equations are dependent and represent the same line. Therefore, there are infinitely many solutions along the line described by \( x = 3y - 1 \).<br /><br />### Problem 8:<br /><br />We have the following system of equations:<br />1. \( 4x - 7y = 7 \)<br />2. \( 12x - 21y = 24 \)<br /><br />**Step 1: Simplify Equation 2**<br /><br />Notice that Equation 2 can be simplified by dividing all terms by 3:<br /><br />\[ \frac{12x}{3} - \frac{21y}{3} = \frac{24}{3} \]<br /><br />This simplifies to:<br /><br />\[ 4x - 7y = 8 \]<br /><br />**Step 2: Compare with Equation 1**<br /><br />Now we have:<br />1. \( 4x - 7y = 7 \)<br />2. \( 4x - 7y = 8 \)<br /><br />These two equations are inconsistent because they imply that the same linear combination of \( x \) and \( y \) equals two different numbers (7 and 8). Therefore, there is no solution to this system; the lines are parallel and do not intersect.<br /><br />In summary:<br />- For Problem 7, there are infinitely many solutions.<br />- For Problem 8, there is no solution.