Problems 15 through 18 are systems whose equations are either dependent or inconsistent. Try to solve Then write a conclu sion about how augmented matrices tell you when you have this kind of system, and how they allow you to distinguish between dependent equal tions and inconsistent equations, 15 10x+15y=21 12x+18y=35 16 30x+48y=129 29x+32y=24 24x+18y=66 28x+21y=77 18 25x+10y=37 29x+8y=51

To determine whether the systems of equations are dependent or inconsistent, we can use augmented matrices. Let's solve each system and analyze the results.<br /><br />### Problem 15<br />System of equations:<br />\[ <br />\begin{cases}<br />10x + 15y = 21 \\<br />12x + 18y = 35<br />\end{cases}<br />\]<br /><br />**Step 1: Form the augmented matrix.**<br />\[<br />\left[\begin{array}{cc|c}<br />10 & 15 & 21 \\<br />12 & 18 & 35<br />\end{array}\right]<br />\]<br /><br />**Step 2: Apply row operations to achieve row-echelon form.**<br /><br />Divide the first row by 10:<br />\[<br />\left[\begin{array}{cc|c}<br />1 & 1.5 & 2.1 \\<br />12 & 18 & 35<br />\end{array}\right]<br />\]<br /><br />Subtract 12 times the first row from the second row:<br />\[<br />\left[\begin{array}{cc|c}<br />1 & 1.5 & 2.1 \\<br />0 & 0 & 0<br />\end{array}\right]<br />\]<br /><br />**Step 3: Analyze the resulting matrix.**<br />The second row is all zeros, indicating that the system has infinitely many solutions. This means the equations are dependent.<br /><br />### Problem 16<br />System of equations:<br />\[ <br />\begin{cases}<br />30x + 48y = 129 \\<br />29x + 32y = 24 \\<br />24x + 18y = 66 \\<br />28x + 21y = 77<br />\end{cases}<br />\]<br /><br />**Step 1: Form the augmented matrix.**<br />\[<br />\left[\begin{array}{cccc|c}<br />30 & 48 & 0 & 0 & 129 \\<br />29 & 32 & 0 & 0 & 24 \\<br />24 & 18 & 0 & 0 & 66 \\<br />28 & 21 & 0 & 0 & 77<br />\end{array}\right]<br />\]<br /><br />**Step 2: Apply row operations to achieve row-echelon form.**<br /><br />Divide the first row by 30:<br />\[<br />\left[\begin{array}{cccc|c}<br />1 & 1.6 & 0 & 0 & 4.3 \\<br />29 & 32 & 0 & 0 & 24 \\<br />24 & 18 & 0 & 0 & 66 \\<br />28 & 21 & 0 & 0 & 77<br />\end{array}\right]<br />\]<br /><br />Subtract 29 times the first row from the second row:<br />\[<br />\left[\begin{array}{cccc|c}<br />1 & 1.6 & 0 & 0 & 4.3 \\<br />0 & -1.04 & 0 & 0 & -18.11 \\<br />24 & 18 & 0 & 0 & 66 \\<br />28 & 21 & 0 & 0 & 77<br />\end{array}\right]<br />\]<br /><br />Subtract 24 times the first row from the third row:<br />\[<br />\left[\begin{array}{cccc|c}<br />1 & 1.6 & 0 & 0 & 4.3 \\<br />0 & -1.04 & 0 & 0 & -18.11 \\<br />0 & -9.36 & 0 & 0 & -49.32 \\<br />28 & 21 & 0 & 0 & 77<br />\end{array}\right]<br />\]<br /><br />Subtract 28 times the first row from the fourth row:<br />\[<br />\left[\begin{array}{cccc|c}<br />1 & 1.6 & 0 & 0 & 4.3 \\<br />0 & -1.04 & 0 & 0 & -18.11 \\<br />0 & -9.36 & 0 & 0 & -49.32 \\<br />0 & -25.28 & 0 & 0 & -60.36<br />\end{array}\right]<br />\]<br /><br />**Step 3: Analyze the resulting matrix.**<br />All rows are linearly dependent, indicating that the system has no solution. This means the equations are inconsistent.<br /><br />### Problem 17<br />System of equations:<br />\[ <br />\begin{cases}<br />24x + 18y = 66 \\<br />28x + 21y = 77<br />\end{cases}<br />\]<br /><br />**Step 1: Form the augmented matrix.**<br />\[<br />\left[\begin{array}{cc|c}<br />24 & 18 & 66 \\<br />28 & 21 & 77<br />\end{array}\right]<br />\]<br /><br />**Step 2: Apply row operations to achieve row-echelon form.**<br /><br />Divide the first row by 6:<br />\[<br />\left[\begin{array}{cc|c}<br />4 & 3 & 11 \\<br />28 & 21