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Chapter 9: Problem 51

Solve each problem using a system of equations in two variables. Unknown Numbers Find two numbers whose squares have a sum of 100 and a difference of 28

Short Answer

Expert verified
The numbers are (8, 6), (-8, 6), (8, -6), and (-8, -6).

Step by step solution

01

- Define the Variables

Let the two numbers be represented by variables, say, \(x\) and \(y\).
02

- Set Up the Equations

We are given two pieces of information:1. The sum of their squares is 100: \(x^2 + y^2 = 100\)2. The difference of their squares is 28: \(x^2 - y^2 = 28\)
03

- Solve the Equations Simultaneously

Add the two equations together to eliminate \(y\):\(x^2 + y^2 + x^2 - y^2 = 100 + 28\)This simplifies to:\(2x^2 = 128\)Then solve for \(x\):\(x^2 = 64\)\(x = 8\) or \(x = -8\)
04

- Find Corresponding y Values

Substitute \(x = 8\) back into one of the original equations to find \(y\):Using \(x^2 + y^2 = 100\):\(64 + y^2 = 100\)\(y^2 = 36\)\(y = 6\) or \(y = -6\)Similarly, for \(x = -8\), the solutions for \(y\) remain \(6\) or \(-6\).
05

- List All Pairs of Solutions

The pairs of numbers are (8, 6), (8, -6), (-8, 6), and (-8, -6).

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Key Concepts

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

two variables
In this exercise, we are dealing with two variables. Two variables imply that there are two unknowns that we need to find. Here, the two unknowns are represented by the variables \(x\) and \(y\). Using variables allows us to write equations that describe the relationships between the unknowns. For example, if you want to express a relationship where the sum of the squares of two numbers must be 100, you can write this as \(x^2 + y^2 = 100\). This is much simpler and clearer than writing it out in words. When tackling problems involving two variables, it is crucial to define what each variable represents clearly at the beginning.
solving equations
Solving equations is a fundamental part of mathematical problem-solving. In this example, we need to solve two equations to find our two unknown numbers. The equations given are \(x^2 + y^2 = 100\) and \(x^2 - y^2 = 28\). To solve these equations, we need strategies like addition or substitution. First, by adding the two equations together, we eliminate one variable: \(x^2 + y^2 + x^2 - y^2 = 100 + 28\). Simplifying, we get \(2x^2 = 128\), and solve for \(x\): \(x^2 = 64\), so \(x = 8\) or \(x = -8\). This process shows how to systematically find the value of one variable from the equations.
squares of numbers
The squares of numbers are important in many mathematical problems, especially in this one. Squaring a number means multiplying the number by itself, i.e., \(x^2 = x \times x\). In this problem, we need to find two numbers whose squares add up to 100. Hence, our equations, \(x^2 + y^2 = 100\) and \(x^2 - y^2 = 28\), revolve around the squares of \(x\) and \(y\). Squaring numbers affects the sign, because the square of both positive and negative numbers will always be positive. This is why for \(x^2 = 64\), \(x\) can be 8 or -8. The same logic applies when solving for \(y\). Understanding how squaring works is essential for solving such equations.
simultaneous equations
Simultaneous equations are a set of equations with multiple variables that are solved together. For this problem, we have two simultaneous equations involving \(x\) and \(y\): \(x^2 + y^2 = 100\) and \(x^2 - y^2 = 28\). The goal is to find values of \(x\) and \(y\) that satisfy both equations at the same time. To solve simultaneous equations, we often use methods such as addition, subtraction, or substitution. For example, adding the two given equations helps eliminate \(y\), making it easier to solve for \(x\). This is why learning how to handle simultaneous equations is crucial for tackling more complex problems in algebra and calculus.
mathematical problem solving
Mathematical problem-solving involves logical thinking and applying various techniques to find a solution. In this exercise, we defined our variables, set up equations, and solved them step-by-step. Here are the steps to solve it effectively:
  • Define your variables clearly.
  • Set up your equations based on the problem statement.
  • Use logical operations to simplify and solve the equations.
  • Verify your solutions by substituting them back into the original equations.
By following these steps, the problem of finding two numbers whose squares add up to 100 and have a difference of 28 becomes manageable. This structured approach can be applied to a wide range of mathematical problems you may encounter.

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Most popular questions from this chapter

Find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points \((2,3),(-1,0),\) and \((-2,2)\) Find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points \((-2,4),(2,2),\) and \((4,9)\)

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} &0.2 x+0.3 y=-1.9\\\ &0.7 x-0.2 y=4.6 \end{aligned}$$

Storage Capacity An office manager wants to buy some filing cabinets. He knows that cabinet A costs \(\$ 10\) each, requires \(6 \mathrm{ft}^{2}\) of floor space, and holds \(8 \mathrm{ft}^{3}\) of files. Cabinet B costs \(\$ 20\) each, requires \(8 \mathrm{ft}^{2}\) of floor space, and holds \(12 \mathrm{ft}^{3}\) of files. He can spend no more than \(\$ 140\) due to budget limitations, and his office has room for no more than \(72 \mathrm{ft}^{2}\) of cabinets. He wants to maximize storage capacity within the limits imposed by funds and space. How many of each type of cabinet should he buy?

Solve each problem. Tire Sales The number of automobile tire sales is dependent on several variables. In one study the relationship among annual tire sales \(S\) (in thousands of dollars), automobile registrations \(R\) (in millions), and personal disposable income \(I\) (in millions of dollars) was investigated. The results for three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b R+c l $$ where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Jarrett, J., Business Forecasting Methods, Basil Blackwell, Ltd.) (a) Substitute the values for \(S, R,\) and \(I\) for each year from the table into the equation \(S=a+b R+c I,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. (d) If \(R=117.6\) and \(I=310.73,\) predict \(S .\) (The actual value for \(S\) was \(11,314 .\) ) (e) If \(R=143.8 \text { and } I=829.06, \text { predict } S . \text { (The actual value for } S \text { was } 18,481 .)\) $$\begin{array}{|c|c|c|} \hline S & R & I \\ \hline 10,170 & 112.9 & 307.5 \\\ \hline 15,305 & 132.9 & 621.63 \\ \hline 21,289 & 155.2 & 1937.13 \\\ \hline \end{array}$$

Solve each system by using the inverse of the coefficient matrix. $$\begin{array}{r} 6 x+9 y=3 \\ -8 x+3 y=6 \end{array}$$
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