Problem 45
Question
Let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The difference between the squares of two numbers is 3 . Twice the square of the first number increased by the square of the second number is 9. Find the numbers.
Step-by-Step Solution
Verified Answer
The two numbers are either 2 and 1, or -2 and -1.
1Step 1: Formulate the equations
From 'The difference between the squares of two numbers is 3', the equation is \(x^2 - y^2 = 3\). From 'Twice the square of the first number increased by the square of the second number is 9', the equation will be \(2x^2 + y^2 = 9\). So, the system of equations to solve is: \n1) \(x^2 - y^2 = 3\) \n2) \(2x^2 + y^2 = 9\)
2Step 2: Solve the system
The first equation can be rearranged to obtain \(x^2 = y^2 + 3\). Substitute this into equation 2 to get: \(2(y^2 + 3) + y^2 = 9\), which simplifies to \(3y^2 + 6 = 9\). Further simplification gives: \(3y^2 = 3\), thereby yielding \(y^2 = 1\), so \(y = \pm 1\). Now, substitute \(y = 1\) and \(y = -1\) separately into equation 1 to find the corresponding x-values.
3Step 3: Find x-values
Substituting \(y = 1\) into equation 1, we get: \(x^2 - 1^2 = 3\), which simplifies to \(x^2 = 4\). Thus, \(x = \pm 2\). On the other hand, substituting \(y = -1\) into equation 1 yields: \(x^2 - (-1)^2 = 3\), simplifying to \(x^2 = 4\) giving \(x = \pm 2\). Therefore, the two numbers are 2 and 1 or -2 and -1.
Key Concepts
Algebraic EquationsQuadratic EquationsSubstitution Method
Algebraic Equations
Algebraic equations are the foundation of many mathematical problems and solving them is a critical skill in mathematics. An algebraic equation is a statement of equality that contains one or more variables. These equations can range from simple linear equations with a single variable to more complex polynomial equations with multiple variables.
In the exercise provided, we're given a scenario that requires forming a system of equations to find two numbers. The equations are based on the relationships between the squares of these numbers. Understanding how to manipulate and solve these algebraic equations is essential to finding the correct solution. Solving algebraic equations often involves combining like terms, using the distributive property, and applying inverse operations to isolate the variable. By mastering these techniques, students can approach a wide variety of algebraic problems with confidence.
In the exercise provided, we're given a scenario that requires forming a system of equations to find two numbers. The equations are based on the relationships between the squares of these numbers. Understanding how to manipulate and solve these algebraic equations is essential to finding the correct solution. Solving algebraic equations often involves combining like terms, using the distributive property, and applying inverse operations to isolate the variable. By mastering these techniques, students can approach a wide variety of algebraic problems with confidence.
Quadratic Equations
Quadratic equations are a specific type of algebraic equation that take the form of ax^2 + bx + c = 0, where a, b, and c are known values, and x is the variable to be solved for. They are characterized by the presence of the x^2 term, which indicates that the variable is squared.
In our exercise, the equations derived from the problem statement are not linear but quadratic in nature, as they involve the square of the unknown numbers. The challenge here involves not only setting up correct equations but also manipulating and solving them. Quadratic equations can have two solutions, corresponding to the fact that a number and its negative counterpart have the same square. Recognizing that quadratics may yield more than one solution is crucial, and using methods such as factoring, completing the square, or the quadratic formula is often necessary to find these solutions. As illustrated in the solution, finding the values of x and y from the quadratic system entails using substitution and simplification to arrive at the possible values for y and, subsequently, for x.
In our exercise, the equations derived from the problem statement are not linear but quadratic in nature, as they involve the square of the unknown numbers. The challenge here involves not only setting up correct equations but also manipulating and solving them. Quadratic equations can have two solutions, corresponding to the fact that a number and its negative counterpart have the same square. Recognizing that quadratics may yield more than one solution is crucial, and using methods such as factoring, completing the square, or the quadratic formula is often necessary to find these solutions. As illustrated in the solution, finding the values of x and y from the quadratic system entails using substitution and simplification to arrive at the possible values for y and, subsequently, for x.
Substitution Method
The substitution method is a powerful technique used to solve systems of equations, particularly when equations are non-linear, as is the case in our exercise. This method involves expressing one variable in terms of another from one equation and then substituting this expression into the other equation.
The step-by-step solution shows the application of the substitution method effectively. After forming the system of equations, one equation is rearranged to express x^2 in terms of y^2. This expression is then substituted into the second equation, simplifying the problem to one with a single variable. Once y is found, it can be back-substituted into the original equations to find the corresponding x values. This method often simplifies the process of solving complex systems of equations by reducing the number of variables to solve for at one time, as clearly demonstrated in the problem's solution. This method can also be used interchangeably with the elimination method in many cases, providing a flexible approach to solving algebraic equations.
The step-by-step solution shows the application of the substitution method effectively. After forming the system of equations, one equation is rearranged to express x^2 in terms of y^2. This expression is then substituted into the second equation, simplifying the problem to one with a single variable. Once y is found, it can be back-substituted into the original equations to find the corresponding x values. This method often simplifies the process of solving complex systems of equations by reducing the number of variables to solve for at one time, as clearly demonstrated in the problem's solution. This method can also be used interchangeably with the elimination method in many cases, providing a flexible approach to solving algebraic equations.
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