Textbook Question
In Exercises 47–52, solve each system by the method of your choice.
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7. Systems of Equations & Matrices
Two Variable Systems of Linear Equations
6:55 minutesProblem 13
Textbook Question
In Exercises 1–18, solve each system by the substitution method.
Verified step by step guidance1
Start with the given system of equations: \(xy = 3\) and \(x^{2} + y^{2} = 10\).
From the first equation \(xy = 3\), solve for one variable in terms of the other. For example, solve for \(y\): \(y = \frac{3}{x}\).
Substitute the expression for \(y\) into the second equation \(x^{2} + y^{2} = 10\). This gives: \(x^{2} + \left(\frac{3}{x}\right)^{2} = 10\).
Simplify the substituted equation: \(x^{2} + \frac{9}{x^{2}} = 10\). To clear the denominator, multiply both sides of the equation by \(x^{2}\), resulting in \(x^{4} + 9 = 10x^{2}\).
Rewrite the equation as a quadratic in terms of \(x^{2}\): \(x^{4} - 10x^{2} + 9 = 0\). Let \(u = x^{2}\), then solve the quadratic equation \(u^{2} - 10u + 9 = 0\) for \(u\).
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Video duration:
6mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is especially useful when one equation is already solved for a variable or can be easily manipulated.
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Solving Nonlinear Systems
Nonlinear systems include equations where variables are raised to powers or multiplied together, such as xy=3 and x² + y²=10. Solving these requires careful algebraic manipulation, often leading to quadratic equations. Understanding how to handle nonlinear terms is essential to find all possible solutions.
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Quadratic Equations
Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. When using substitution, the system often reduces to a quadratic equation, which can be solved using factoring, completing the square, or the quadratic formula. Recognizing and solving quadratics is key to finding the values of variables.
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