- Substitute the linear equation \(y = x - 3\) into the quadratic equation \(x^2 - 5y = 11\).
- Simplify the resulting expression to obtain the quadratic equation \(x^2 - 5x + 4 = 0\).
- Solve the quadratic by factoring to find the roots \(x = 1\) and \(x = 4\).
- Calculate the corresponding \(y\)-value for \(x = 1\) using \(y = x - 3\), resulting in \(y = -2\).
- The other solution set is \(\boxed{x=1, y=-2}\).
Explanation
- Analyzing the System of Equations
The problem asks us to find the second solution set for the given system of two simultaneous equations:
\[x^2 - 5y = 11\]
\[y = x - 3\]
We are already given one solution: \((4, 1)\). To find the other solution, we will substitute the expression for \(y\) from the second equation into the first equation. This will allow us to solve for \(x\).
- Substituting and Simplifying
Substituting the expression \(y = x - 3\) into the first equation \(x^2 - 5y = 11\), we get:
\[x^2 - 5(x - 3) = 11\]
Expanding the equation:
\[x^2 - 5x + 15 = 11\]
Subtract 11 from both sides to form a quadratic equation:
\[x^2 - 5x + 15 - 11 = 0\]
\[x^2 - 5x + 4 = 0\]
- Solving for the Second Root
We now factor the quadratic equation \(x^2 - 5x + 4 = 0\). We look for two numbers that multiply to \(4\) and add to \(-5\). These numbers are \(-1\) and \(-4\). Thus:
\[(x - 1)(x - 4) = 0\]
The roots of the equation are \(x = 1\) and \(x = 4\). We were already given the solution \(x = 4\), which corresponds to \(y = 4 - 3 = 1\). Now we evaluate the solution for \(x = 1\). Substituting \(x = 1\) back into the second equation \(y = x - 3\):
\[y = 1 - 3\]
\[y = -2\]
- Conclusion and Verification
We have identified the other root as \(x = 1\) with the corresponding value \(y = -2\). Let's verify this solution in the original equations:
Equation 1: \(x^2 - 5y = (1)^2 - 5(-2) = 1 + 10 = 11\). This holds true.
Equation 2: \(y = x - 3 = 1 - 3 = -2\). This also holds true.
Thus, the other solution set is \(x = 1, y = -2\). The final answer is \(\boxed{x=1, y=-2}\).
Examples
Systems of equations are used extensively in economics and finance. For instance, if you represent the supply curve and demand curve for a product as two equations in terms of price (\(p\)) and quantity (\(q\)), the point where both equations are satisfied represents the equilibrium price and quantity. Solving such systems helps businesses and governments determine market stability and set prices efficiently. In our case, finding both solutions allows us to identify the two points where the parabola \(x^2 - 5y = 11\) intersects the line \(y = x - 3\).