How many integral values of x and y satisfy the equation 5x + 9y = 7, where -500 < x < 500 and -500 < y < 500?
111
Solving Linear Diophantine Equations
The problem asks us to find the number of integral solutions \((x, y)\) for the linear equation \(5x + 9y = 7\) that fall within the given ranges for \(x\) and \(y\): \(-500 < x < 500\) and \(-500 < y < 500\). This type of equation, where we are looking for integer solutions, is called a linear Diophantine equation.
Finding a Particular Solution
First, we need to find one specific integral solution \((x_0, y_0)\) to the equation \(5x + 9y = 7\). We can use the Extended Euclidean Algorithm to find integers \(u\) and \(v\) such that \(5u + 9v = \gcd(5, 9)\). The greatest common divisor of 5 and 9 is 1. Since 1 divides 7, we know that integral solutions exist.
Let's find one pair \((u, v)\) for \(5u + 9v = 1\). \(9 = 1 \cdot 5 + 4\) \(5 = 1 \cdot 4 + 1\) \(4 = 4 \cdot 1 + 0\)
Now work backwards: \(1 = 5 - 1 \cdot 4\) Substitute \(4 = 9 - 1 \cdot 5\): \(1 = 5 - 1 \cdot (9 - 1 \cdot 5)\) \(1 = 5 - 9 + 1 \cdot 5\) \(1 = 2 \cdot 5 - 1 \cdot 9\)
So, \(5(2) + 9(-1) = 1\). We want the equation to equal 7, so we multiply this equation by 7: \(7 \cdot (5(2) + 9(-1)) = 7 \cdot 1\) \(5(14) + 9(-7) = 7\)
Thus, a particular solution is \((x_0, y_0) = (14, -7)\).
Finding the General Solution
For a linear Diophantine equation \(ax + by = c\), if \((x_0, y_0)\) is a particular solution and \(d = \gcd(a, b)\), the general solution is given by: \(x = x_0 + (b/d)n\) \(y = y_0 - (a/d)n\) where \(n\) is any integer (\(n \in \mathbb{Z}\)).
In our equation, \(a=5\), \(b=9\), \(c=7\), and \(d=\gcd(5, 9)=1\). Using our particular solution \((x_0, y_0) = (14, -7)\), the general solution is:
\(x = 14 + (9/1)n = 14 + 9n\) \(y = -7 - (5/1)n = -7 - 5n\)
Here, \(n\) can be any integer.
Applying the Constraints
We are given constraints on the values of \(x\) and \(y\): \(-500 < x < 500\) \(-500 < y < 500\)
Let's apply the first constraint to the expression for \(x\) in terms of \(n\):
\( -500 < 14 + 9n < 500 \\ -500 - 14 < 9n < 500 - 14 \\ -514 < 9n < 486 \)
Now, divide by 9:
\( -514/9 < n < 486/9 \\ -57.11... < n < 54 \)
Since \(n\) must be an integer, this means \(-57 \le n \le 53\).
Let's apply the second constraint to the expression for \(y\) in terms of \(n\):
\( -500 < -7 - 5n < 500 \\ -500 + 7 < -5n < 500 + 7 \\ -493 < -5n < 507 \)
Now, divide by -5. Remember to reverse the inequality signs when dividing by a negative number:
\( -493/(-5) > n > 507/(-5) \\ 98.6 > n > -101.4 \)
Rearranging the inequality: \(-101.4 < n < 98.6\)
Since \(n\) must be an integer, this means \(-101 \le n \le 98\).
Finding the Common Range for n
The integral values of \(n\) must satisfy both conditions simultaneously:
Condition 1: \(-57 \le n \le 53\) Condition 2: \(-101 \le n \le 98\)
To satisfy both, \(n\) must be greater than or equal to the maximum of the lower bounds and less than or equal to the minimum of the upper bounds.
Lower bound for \(n\): \(\max(-57, -101) = -57\). Upper bound for \(n\): \(\min(53, 98) = 53\).
So, the integral values of \(n\) that satisfy both constraints are \(-57 \le n \le 53\).
Counting the Number of Solutions
To find the number of integral values of \(n\) in the range \([-57, 53]\), we use the formula: Number of values = (Largest value) - (Smallest value) + 1.
Number of values of \(n = 53 - (-57) + 1 = 53 + 57 + 1 = 110 + 1 = 111\).
Each integer value of \(n\) corresponds to a unique integral solution \((x, y)\) that satisfies the given equation and the specified ranges for \(x\) and \(y\). Therefore, there are 111 integral solutions.
| Step | Description | Result |
|---|---|---|
| 1 | Find GCD(5, 9) | 1 |
| 2 | Find a particular solution \((x_0, y_0)\) for \(5x + 9y = 7\) | \((14, -7)\) |
| 3 | Write the general solution for \((x, y)\) in terms of \(n\) | \(x = 14 + 9n\), \(y = -7 - 5n\) |
| 4 | Apply constraint \(-500 < x < 500\) to find range of \(n\) | \(-57 \le n \le 53\) |
| 5 | Apply constraint \(-500 < y < 500\) to find range of \(n\) | \(-101 \le n \le 98\) |
| 6 | Find the common range for \(n\) satisfying both constraints | \(-57 \le n \le 53\) |
| 7 | Count the number of integers in the common range | \(53 - (-57) + 1 = 111\) |
Integral Solutions and Constraints
We found that the problem of counting integral values of \(x\) and \(y\) satisfying \(5x + 9y = 7\) within the given bounds transforms into counting the number of integers \(n\) that satisfy derived inequalities. The ranges for \(x\) and \(y\) restrict the possible values of \(n\). The key was to find the intersection of the allowed ranges for \(n\) from both constraints.
Revision Table: Diophantine Equation Solutions
| Concept | Explanation | Relevance Here |
|---|---|---|
| Linear Diophantine Equation | An equation of the form \(ax + by = c\) where \(a, b, c\) are integers and we seek integer solutions for \(x\) and \(y\). | The given equation \(5x + 9y = 7\) is a linear Diophantine equation. |
| Existence of Solutions | Integral solutions exist if and only if \(\gcd(a, b)\) divides \(c\). | \(\gcd(5, 9)=1\), and 1 divides 7, so solutions exist. |
| Particular Solution | Any single integer pair \((x_0, y_0)\) that satisfies the equation \(ax + by = c\). Can be found using Extended Euclidean Algorithm or trial and error. | We found \((14, -7)\) as a particular solution. |
| General Solution | A formula that gives all possible integer solutions based on a particular solution \((x_0, y_0)\) and \(d = \gcd(a, b)\): \(x = x_0 + (b/d)n\), \(y = y_0 - (a/d)n\) for \(n \in \mathbb{Z}\). | Used to express \(x\) and \(y\) in terms of a single integer parameter \(n\). |
| Applying Inequalities | Substitute the general solution expressions into the given constraints (inequalities) on \(x\) and \(y\). | This step yielded inequalities for \(n\) based on the bounds for \(x\) and \(y\). |
| Finding Common Range | Determine the range of the integer parameter \(n\) that satisfies all derived inequalities simultaneously. | The intersection of the ranges for \(n\) from both constraints gives the valid range for \(n\). |
| Counting Integers | Count the number of integers within a given range \([a, b]\) using the formula \(b - a + 1\). | Used to find the total number of valid values for \(n\), which equals the number of integral solutions \((x, y)\). |
Additional Information: Variations and Other Methods
While we used a particular solution derived from the Euclidean algorithm, one could also find a particular solution by inspection. For \(5x + 9y = 7\), we might notice that \(5 \cdot 2 + 9 \cdot (-1/3)\) is not integer, but \(5 \cdot 5 + 9 \cdot (-2) = 25 - 18 = 7\). So \((5, -2)\) is another particular solution. If we use \((5, -2)\) as \((x_0, y_0)\), the general solution becomes \(x = 5 + 9n\) and \(y = -2 - 5n\). Let's quickly check the constraints with this general form:
Constraint on \(x\): \(-500 < 5 + 9n < 500 \implies -505 < 9n < 495 \implies -56.11... < n < 55 \implies -56 \le n \le 54\).
Constraint on \(y\): \(-500 < -2 - 5n < 500 \implies -498 < -5n < 502 \implies -100.4 < n < 99.6 \implies -100 \le n \le 99\).
The common range for \(n\) is \(\max(-56, -100) \le n \le \min(54, 99)\), which is \(-56 \le n \le 54\).
Number of values of \(n = 54 - (-56) + 1 = 54 + 56 + 1 = 111\).
As expected, the number of solutions is the same regardless of the particular solution used, as long as the general solution formula is applied correctly. The parameter \(n\) simply shifts its range, but the number of integers within that range remains constant.
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