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Chapter 10: Problem 14

Use the elimination method to find all solutions of the system of equations. $$\left\\{\begin{array}{c} x^{2}-y^{2}=1 \\ 2 x^{2}-y^{2}=x+3 \end{array}\right.$$

Short Answer

Expert verified
The solutions are \((2, \\pm \\sqrt{3})\) and \((-1, 0)\).

Step by step solution

01

Examine the given equations

The system of equations given is \(x^2 - y^2 = 1\) and \(2x^2 - y^2 = x + 3\). Our goal is to eliminate one of the variables to solve the system.
02

Eliminate \(y^2\) by subtraction

Subtract the first equation from the second equation to eliminate \(y^2\).\[ (2x^2 - y^2) - (x^2 - y^2) = (x + 3) - 1 \]This simplifies to:\[ x^2 = x + 2 \]
03

Rearrange and solve the quadratic equation

Rearrange the equation \(x^2 = x + 2\) to form a quadratic equation:\[ x^2 - x - 2 = 0 \]Use factoring to solve it:\( x^2 - x - 2 = (x - 2)(x + 1) = 0 \)Thus, the solutions for \(x\) are \(x = 2\) and \(x = -1\).
04

Solve for \(y\) using \(x = 2\)

Substitute \(x = 2\) into the first equation:\(x^2 - y^2 = 1\) becomes \(4 - y^2 = 1\).Solve for \(y^2\):\[ y^2 = 4 - 1 = 3 \]Thus, \(y = \pm \sqrt{3}\). Hence, the solutions for \(x = 2\) are \((2, \sqrt{3})\) and \((2, -\sqrt{3})\).
05

Solve for \(y\) using \(x = -1\)

Substitute \(x = -1\) into the first equation:\(x^2 - y^2 = 1\) becomes \(1 - y^2 = 1\).Solve for \(y^2\):\[ y^2 = 1 - 1 = 0 \]Thus, \(y = 0\). Hence, the solution for \(x = -1\) is \((-1, 0)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Equations
A system of equations consists of two or more equations with the same set of variables. In our example, we are dealing with a system of two equations involving the variables \(x\) and \(y\).
  • The first equation: \(x^2 - y^2 = 1\)
  • The second equation: \(2x^2 - y^2 = x + 3\)
The goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously. There are various methods to solve such a system, including substitution, graphing, and elimination. In this particular example, we utilize the elimination method.

The elimination method involves combining equations to cancel out one of the variables. This simplifies the problem from a system of equations to a single equation. Once one variable is eliminated, it becomes easier to solve for the remaining variable.
Quadratic Equation
A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\). The quadratic equation featured in this exercise is derived after manipulating the system of equations:

\[ x^2 = x + 2 \]
Rearranging gives us:
\[ x^2 - x - 2 = 0 \]
To solve quadratic equations, we commonly use methods such as:
  • Factoring, if possible
  • Using the quadratic formula
  • Completing the square
For our exercise, the equation is easily factorable, simplifying the process of finding the solutions. It reveals the possible values of \(x\), which we then use to find the corresponding \(y\) values.
Factoring
Factoring involves expressing a polynomial as a product of its factors. In our quadratic equation \(x^2 - x - 2 = 0\), we use the factoring method to make solving for \(x\) straightforward. Let's see how:

First, identify two numbers that multiply to the constant term, which is \(-2\), and add to the linear coefficient, which is \(-1\). These numbers are \(-2\) and \(+1\).

The equation can then be factored as:
\( (x - 2)(x + 1) = 0 \)
Setting each factor equal to zero gives the solutions:
  • \(x - 2 = 0\) results in \(x = 2\)
  • \(x + 1 = 0\) results in \(x = -1\)
Factoring is a powerful tool since it decomposes complex equations into simpler, more manageable parts that are easier to solve. With these \(x\) values, finding the corresponding \(y\) in the system becomes easily achievable.

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Most popular questions from this chapter

A publishing company publishes a total of no more than 100 books every year. At least 20 of these are nonfiction, but the company always publishes at least as much fiction as nonfiction. Find a system of inequalities that describes the possible numbers of fiction and nonfiction books that the company can produce each year consistent with these policies. Graph the solution set.

(a) If three points lie on a line, what is the area of the "triangle" that they determine? Use the answer to this question, together with the determinant formula for the area of a triangle, to explain why the points \(\left(a_{1}, b_{1}\right)\) \(\left(a_{2}, b_{2}\right),\) and \(\left(a_{3}, b_{3}\right)\) are collinear if and only if $$\left|\begin{array}{lll} a_{1} & b_{1} & 1 \\ a_{2} & b_{2} & 1 \\ a_{3} & b_{3} & 1 \end{array}\right|=0$$ (b) Use a determinant to check whether each set of points is collinear. Graph them to verify your answer. (I) \((-6,4),(2,10),(6,13)\) (II) \((-5,10),(2,6),(15,-2)\)

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{c} y>x+1 \\ x+2 y \leq 12 \\ x+1>0 \end{array}\right.$$

Nutrition A doctor recommends that a patient take \(50 \mathrm{mg}\) each of niacin, riboflavin, and thiamin daily to alleviate a vitamin deficiency. In his medicine chest at home the patient finds three brands of vitamin pills. The amounts of the relevant vitamins per pill are given in the table. How many pills of each type should he take every day to get \(50 \mathrm{mg}\) of each vitamin? $$\begin{array}{|l|c|c|c|} \hline & \text { VitaMax } & \text { Vitron } & \text { VitaPlus } \\ \hline \text { Niacin (mg) } & 5 & 10 & 15 \\ \text { Riboflavin (mg) } & 15 & 20 & 0 \\ \text { Thiamin (mg) } & 10 & 10 & 10 \\ \hline \end{array}$$

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} y & \geq x^{2} \\ x+y & \geq 6 \end{aligned}\right.$$
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