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Chapter 1: Problem 18

Find all solutions in nonnegative integers to the system of equations $$ 3 x^2-2 y^2-4 z^2+54=0, \quad 5 x^2-3 y^2-7 z^2+74=0 $$

Short Answer

Expert verified
The nonnegative integer solution is \(x = 4\), \(y = 7\), and \(z = 1\).

Step by step solution

01

Write the given equations

We are given the system of equations: 1) \(3x^2 - 2y^2 - 4z^2 + 54 = 0\) 2) \(5x^2 - 3y^2 - 7z^2 + 74 = 0\).
02

Eliminate one variable

To simplify, we can eliminate one variable by multiplying the first equation by 5 and the second equation by 3 to get a common coefficient for \(x^2\): 1) \(15x^2 - 10y^2 - 20z^2 + 270 = 0\) 2) \(15x^2 - 9y^2 - 21z^2 + 222 = 0\).
03

Subtract equations

Subtract the modified second equation from the modified first equation to eliminate \(x^2\): \((15x^2 - 10y^2 - 20z^2 + 270) - (15x^2 - 9y^2 - 21z^2 + 222) = 0\), which simplifies to: \(-y^2 + z^2 + 48 = 0\).
04

Solve for one of the variables

Rearrange the equation from Step 3 to solve for \(y^2\): \(y^2 = z^2 + 48\).
05

Substitute back into one of the original equations

Substitute \(y^2 = z^2 + 48\) back into the first original equation, \(3x^2 - 2(y^2) - 4z^2 + 54 = 0\): \(3x^2 - 2(z^2 + 48) - 4z^2 + 54 = 0\).
06

Simplify and solve for the variables

Simplify the equation: \(3x^2 - 2z^2 - 96 - 4z^2 + 54 = 0\) \(3x^2 - 6z^2 - 42 = 0\) \(3x^2 = 6z^2 + 42\) \(x^2 = 2z^2 + 14\). Since we need nonnegative integer solutions, we test possible values for \(z\) (\(z = 0, 1, 2\), etc.) and solve for \(x\) each time.
07

Check for each possible value of \(z\)

For \(z = 0\): \(x^2 = 2(0)^2 + 14\) \(x^2 = 14\) Since 14 is not a perfect square, this is not a solution. For \(z = 1\): \(x^2 = 2(1)^2 + 14\) \(x^2 = 16\) \(x = 4\). Substitute \(x = 4\), \(z = 1\), and solve for \(y\): \(y^2 = z^2 + 48\) \(y^2 = 1 + 48\) \(y = 7\).
08

Verify the solution

Verify by substituting \(x = 4\), \(y = 7\), and \(z = 1\) back into the original equations: 1) \(3(4)^2 - 2(7)^2 - 4(1)^2 + 54 = 0\) \(3(16) - 2(49) - 4 + 54 = 0\) \(48 - 98 - 4 + 54 = 0\) \(0 = 0\). 2) \(5(4)^2 - 3(7)^2 - 7(1)^2 + 74 = 0\) \(5(16) - 3(49) - 7 + 74 = 0\) \(80 - 147 - 7 + 74 = 0\) \(0 = 0\). The solution is verified as correct.

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Key Concepts

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

Nonnegative Integers
Nonnegative integers refer to whole numbers that are zero or positive. This includes 0, 1, 2, and so on. For example, in this exercise, we are asked to find all solutions in nonnegative integers for the system of equations. This means that the values we find for the variables x, y, and z should be nonnegative.
It鈥檚 crucial to understand this because the allowed values are limited to 0 or positive whole numbers. When solving the equations, you must ensure that none of the variables take on negative or fractional values. Keeping this restriction in mind helps you quickly eliminate some potential solutions.
Elimination Method
The elimination method, also known as the addition method, is a technique used to solve systems of linear equations. By adding or subtracting equations, we eliminate one of the variables, making it easier to solve for the remaining variables.

In this exercise, the elimination method is used to eliminate one of the variables from the system of equations. We multiplied each equation to get a common coefficient for the variable x虏 and then subtracted one equation from the other. This allowed us to simplify the system into a manageable equation with fewer variables:
  • Multiply the first equation by 5 and the second by 3 to align the coefficients of x虏.
  • Subtract the modified second equation from the modified first equation to eliminate x虏.
Learning the elimination method is essential because it simplifies complex systems, making them easier to solve. Once reduced, you can solve for one variable and substitute back to find the others.
Verification of Solutions
Verifying solutions is an important step to ensure that the calculated values satisfy the original equations. In this problem, after determining potential solutions, we substitute these values back into the original equations to verify their correctness.

For example, let's check if the solutions x=4, y=7, and z=1 are correct:
  • Substitute x, y, and z into the first original equation: 3(4)虏 - 2(7)虏 - 4(1)虏 + 54 = 0. Simplified: 48 - 98 - 4 + 54 = 0, which holds true as 0 = 0.
  • Do the same for the second original equation: 5(4)虏 - 3(7)虏 - 7(1)虏 + 74 = 0. Simplified: 80 - 147 - 7 + 74 = 0, which is again true as 0 = 0.
Checking the solutions prevents mistakes in calculation and ensures the correctness of your results. This step is like a final

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Most popular questions from this chapter

Consider the line segments in the \(x y\)-plane formed by connecting points on the positive \(x\)-axis with \(x\) an integer to points on the positive \(y\)-axis with \(y\) an integer. We call a point in the first quadrant an \(I\)-point if it is the intersection of two such line segments. We call a point an \(L\)-point if there is a sequence of distinct \(I\)-points whose limit is the given point. Prove or disprove: If \((x, y)\) is an \(L\)-point, then either \(x\) or \(y\) (or both) is an integer.

Note that if we tile the plane with black and white squares in a regular "checkerboard" pattern, then every square has an equal number of black and of white neighbors (four each), where two squares are considered neighbors if they are not the same but they have at least one common point. If we try the analogous pattern of cubes in 3-space, it no longer works this way: every white cube has 14 black neighbors and only 12 white neighbors, and vice versa. a. Show that there is a different color pattern of black and white "grid" cubes in 3-space for which every cube does have exactly 13 neighbors of each color. b. What happens in \(n\)-space for \(n>3\) ? Is it still possible to find a color pattern for a regular grid of "hypercubes" so that every hypercube, whether black or white, has an equal number of black and white neighbors? If so, show why; if not, give an example of a specific \(n\) for which it is impossible.

Let \(S\) be a set of numbers which includes the elements 0 and 1 . Suppose \(S\) has the property that for any nonempty finite subset \(T\) of \(S\), the average of all the numbers in \(T\) is an element of \(S\). Prove or disprove: \(S\) must contain all the rational numbers between 0 and 1 .

a. If a rational function (a quotient of two real polynomials) takes on rational values for infinitely many rational numbers, prove that it may be expressed as the quotient of two polynomials with rational coefficients. b. If a rational function takes on integral values for infinitely many integers, prove that it must be a polynomial with rational coefficients.

As you might expect, ice fishing is a popular "outdoor" pastime during the long Wohascum County winters. Recently two ice fishermen arrived at Round Lake, which is perfectly circular, and set up their ice houses in exactly opposite directions from the center, two-thirds of the way from the center to the lakeshore. The point of this symmetrical arrangement was that any fish that could be lured would (perhaps) swim toward the closest lure, so that both fishermen would have equal expectations of their catch. Some time later, a third fisherman showed up, and since the first two adamantly refused to move their ice houses, the following problem arose. Could a third ice house be put on the lake in such a way that all three fishermen would have equal expectations at least to the extent that the three regions, each consisting of all points on the lake for which one of the three ice houses was closest, would all have the same area?
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