Solving Systems of Equations to Find Ordered Pairs

Understanding and solving systems of equations is a fundamental concept in algebra. In this article, we will explore a step-by-step method to find the ordered pairs that satisfy a given system of equations. Specifically, we will work through the problem:

Given:

x 2y 5 y 2x - 3x 9 → Simplified as y 2x - 9

Step-by-Step Solution

Substitute the expression for (y) from the first equation into the second equation:

From the first equation, we can express y in terms of x as:
y (frac{x - 5}{2})

Now substitute this expression for (y) into the second simplified equation:

(frac{x - 5}{2}) 2x - 9)

Eliminate the fraction:

Multiplying both sides of the equation by 2 to eliminate the fraction:

x - 5 4x - 18

Expand the right side:

4x - 9 4x^2 - 15x - 27

Rearranging the equation:

4x^2 - 29x 49 0

Solve the quadratic equation:

Using the quadratic formula, where a 4, b -29, c 49:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Calculate the discriminant:

b^2 - 4ac 841 - 1960 -1625

Now we can find the solutions for (x):

x frac{29 pm sqrt{1625}}{8}

Simplify and solve for (x):

sqrt{1625} sqrt{25 cdot 65} 5sqrt{65}

x (frac{29 pm 5sqrt{65}}{8})

Therefore, we have two distinct values for (x):

x1 (frac{29 5sqrt{65}}{8}), and x2 (frac{29 - 5sqrt{65}}{8})

Find corresponding (y) values:

For each (x) value, we can find the corresponding (y) value using the equation (frac{x - 5}{2}).

For x1 (frac{29 5sqrt{65}}{8}) and x2 (frac{29 - 5sqrt{65}}{8}),

y1 (frac{frac{29 5sqrt{65}}{8} - 5}{2}) and y2 (frac{frac{29 - 5sqrt{65}}{8} - 5}{2})

Therefore, the ordered pairs that satisfy the system of equations are:

(x1, y1) (left(frac{29 5sqrt{65}}{8}, frac{frac{29 5sqrt{65}}{8} - 5}{2}right)) (x2, y2) (left(frac{29 - 5sqrt{65}}{8}, frac{frac{29 - 5sqrt{65}}{8} - 5}{2}right))

Conclusion

The number of ordered pairs that satisfy the system of equations is: 2.