9. The sum of two numbers is 8 and their product is -33 find the two numbers 10 A bill of 135 naira was paid using 5 naira note and 10 naira note. If 16 notes were used altogether, find howmany of each were used.

Let's solve the given problems step by step.<br /><br />### Problem 9: Finding Two Numbers<br /><br />Given:<br />- The sum of two numbers is 8.<br />- The product of the two numbers is -33.<br /><br />Let's denote the two numbers as \( x \) and \( y \).<br /><br />We have the following equations:<br />1. \( x + y = 8 \)<br />2. \( xy = -33 \)<br /><br />To find the values of \( x \) and \( y \), we can solve these equations simultaneously.<br /><br />First, let's express \( y \) in terms of \( x \) using the first equation:<br />\[ y = 8 - x \]<br /><br />Now, substitute this expression for \( y \) into the second equation:<br />\[ x(8 - x) = -33 \]<br />\[ 8x - x^2 = -33 \]<br />\[ x^2 - 8x - 33 = 0 \]<br /><br />This is a quadratic equation. We can solve it using the quadratic formula:<br />\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]<br /><br />Here, \( a = 1 \), \( b = -8 \), and \( c = -33 \):<br />\[ x = \frac{8 \pm \sqrt{64 + 132}}{2} \]<br />\[ x = \frac{8 \pm \sqrt{196}}{2} \]<br />\[ x = \frac{8 \pm 14}{2} \]<br /><br />So, the solutions are:<br />\[ x = \frac{22}{2} = 11 \]<br />\[ x = \frac{-6}{2} = -3 \]<br /><br />Therefore, the two numbers are \( 11 \) and \( -3 \).<br /><br />### Problem 10: Finding the Number of 5 Naira and 10 Naira Notes<br /><br />Given:<br />- A bill of 135 naira was paid using 5 naira notes and 10 naira notes.<br />- A total of 16 notes were used.<br /><br />Let's denote the number of 5 naira notes as \( x \) and the number of 10 naira notes as \( y \).<br /><br />We have the following equations:<br />1. \( 5x + 10y = 135 \)<br />2. \( x + y = 16 \)<br /><br />First, let's simplify the first equation by dividing everything by 5:<br />\[ x + 2y = 27 \]<br /><br />Now we have the system of linear equations:<br />1. \( x + 2y = 27 \)<br />2. \( x + y = 16 \)<br /><br />We can solve this system by elimination. Subtract the second equation from the first:<br />\[ (x + 2y) - (x + y) = 27 - 16 \]<br />\[ x + 2y - x - y = 11 \]<br />\[ y = 11 \]<br /><br />Now substitute \( y = 11 \) back into the second equation:<br />\[ x + 11 = 16 \]<br />\[ x = 5 \]<br /><br />Therefore, the number of 5 naira notes used is \( 5 \) and the number of 10 naira notes used is \( 11 \).