Answer :
Sure, let's solve the given system of equations step by step:
Given the equations:
1. [tex]\( x - y = 7 \)[/tex] [tex]\((\text{1})\)[/tex]
2. [tex]\( xy = 15 \)[/tex] [tex]\((\text{2})\)[/tex]
Our task is to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Step 1: Solve equation (1) for [tex]\( x \)[/tex]
[tex]\[ x = y + 7 \][/tex]
Step 2: Substitute this value of [tex]\( x \)[/tex] into equation (2)
[tex]\[ (y + 7)y = 15 \][/tex]
[tex]\[ y^2 + 7y - 15 = 0 \][/tex]
Now we have a quadratic equation:
[tex]\[ y^2 + 7y - 15 = 0 \][/tex]
Step 3: Solve the quadratic equation for [tex]\( y \)[/tex]
To solve for [tex]\( y \)[/tex], we use the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = -15 \)[/tex].
[tex]\[ y = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 1 \cdot (-15)}}{2 \cdot 1} \][/tex]
[tex]\[ y = \frac{-7 \pm \sqrt{49 + 60}}{2} \][/tex]
[tex]\[ y = \frac{-7 \pm \sqrt{109}}{2} \][/tex]
Thus, we get two values for [tex]\( y \)[/tex]:
[tex]\[ y_1 = \frac{-7 - \sqrt{109}}{2} \][/tex]
[tex]\[ y_2 = \frac{-7 + \sqrt{109}}{2} \][/tex]
Step 4: Find the corresponding [tex]\( x \)[/tex] values
Substitute [tex]\( y_1 \)[/tex] and [tex]\( y_2 \)[/tex] back into [tex]\( x = y + 7 \)[/tex]:
For [tex]\( y_1 = \frac{-7 - \sqrt{109}}{2} \)[/tex]:
[tex]\[ x_1 = \left(\frac{-7 - \sqrt{109}}{2}\right) + 7 \][/tex]
[tex]\[ x_1 = \frac{-7 - \sqrt{109} + 14}{2} \][/tex]
[tex]\[ x_1 = \frac{7 - \sqrt{109}}{2} \][/tex]
For [tex]\( y_2 = \frac{-7 + \sqrt{109}}{2} \)[/tex]:
[tex]\[ x_2 = \left(\frac{-7 + \sqrt{109}}{2}\right) + 7 \][/tex]
[tex]\[ x_2 = \frac{-7 + \sqrt{109} + 14}{2} \][/tex]
[tex]\[ x_2 = \frac{7 + \sqrt{109}}{2} \][/tex]
Step 5: Write down the solutions
The solutions to the system of equations are:
[tex]\[ (x_1, y_1) = \left(\frac{7 - \sqrt{109}}{2}, \frac{-7 - \sqrt{109}}{2}\right) \][/tex]
[tex]\[ (x_2, y_2) = \left(\frac{7 + \sqrt{109}}{2}, \frac{-7 + \sqrt{109}}{2}\right) \][/tex]
Therefore, the set of solutions is:
[tex]\[ \boxed{\left( \frac{7 - \sqrt{109}}{2}, \frac{-7 - \sqrt{109}}{2} \right), \left( \frac{7 + \sqrt{109}}{2}, \frac{-7 + \sqrt{109}}{2} \right)} \][/tex]
Given the equations:
1. [tex]\( x - y = 7 \)[/tex] [tex]\((\text{1})\)[/tex]
2. [tex]\( xy = 15 \)[/tex] [tex]\((\text{2})\)[/tex]
Our task is to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Step 1: Solve equation (1) for [tex]\( x \)[/tex]
[tex]\[ x = y + 7 \][/tex]
Step 2: Substitute this value of [tex]\( x \)[/tex] into equation (2)
[tex]\[ (y + 7)y = 15 \][/tex]
[tex]\[ y^2 + 7y - 15 = 0 \][/tex]
Now we have a quadratic equation:
[tex]\[ y^2 + 7y - 15 = 0 \][/tex]
Step 3: Solve the quadratic equation for [tex]\( y \)[/tex]
To solve for [tex]\( y \)[/tex], we use the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = -15 \)[/tex].
[tex]\[ y = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 1 \cdot (-15)}}{2 \cdot 1} \][/tex]
[tex]\[ y = \frac{-7 \pm \sqrt{49 + 60}}{2} \][/tex]
[tex]\[ y = \frac{-7 \pm \sqrt{109}}{2} \][/tex]
Thus, we get two values for [tex]\( y \)[/tex]:
[tex]\[ y_1 = \frac{-7 - \sqrt{109}}{2} \][/tex]
[tex]\[ y_2 = \frac{-7 + \sqrt{109}}{2} \][/tex]
Step 4: Find the corresponding [tex]\( x \)[/tex] values
Substitute [tex]\( y_1 \)[/tex] and [tex]\( y_2 \)[/tex] back into [tex]\( x = y + 7 \)[/tex]:
For [tex]\( y_1 = \frac{-7 - \sqrt{109}}{2} \)[/tex]:
[tex]\[ x_1 = \left(\frac{-7 - \sqrt{109}}{2}\right) + 7 \][/tex]
[tex]\[ x_1 = \frac{-7 - \sqrt{109} + 14}{2} \][/tex]
[tex]\[ x_1 = \frac{7 - \sqrt{109}}{2} \][/tex]
For [tex]\( y_2 = \frac{-7 + \sqrt{109}}{2} \)[/tex]:
[tex]\[ x_2 = \left(\frac{-7 + \sqrt{109}}{2}\right) + 7 \][/tex]
[tex]\[ x_2 = \frac{-7 + \sqrt{109} + 14}{2} \][/tex]
[tex]\[ x_2 = \frac{7 + \sqrt{109}}{2} \][/tex]
Step 5: Write down the solutions
The solutions to the system of equations are:
[tex]\[ (x_1, y_1) = \left(\frac{7 - \sqrt{109}}{2}, \frac{-7 - \sqrt{109}}{2}\right) \][/tex]
[tex]\[ (x_2, y_2) = \left(\frac{7 + \sqrt{109}}{2}, \frac{-7 + \sqrt{109}}{2}\right) \][/tex]
Therefore, the set of solutions is:
[tex]\[ \boxed{\left( \frac{7 - \sqrt{109}}{2}, \frac{-7 - \sqrt{109}}{2} \right), \left( \frac{7 + \sqrt{109}}{2}, \frac{-7 + \sqrt{109}}{2} \right)} \][/tex]
