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Chapter 5: Problem 41

The difference of two positive numbers is 2 and the difference of their squares is \(44 .\) Find the numbers.

Short Answer

Expert verified
The numbers are 12 and 10.

Step by step solution

01

Identify Given Information

We know the difference between two numbers is 2 and the difference of their squares is 44. Let's denote the two numbers as \(x\) and \(y\). We have: \(x - y = 2\) and \(x^2 - y^2 = 44\).
02

Express the Difference of Squares

Recall the difference of squares formula: \(x^2 - y^2 = (x + y)(x - y)\). Substitute the given difference \(x - y = 2\) into the formula: \(44 = (x + y)(2)\).
03

Solve for \((x + y)\)

Divide both sides of the equation \(44 = 2(x + y)\) by 2 to solve for \(x + y\): \[ x + y = 22. \]
04

Form a System of Equations

We now have a system of linear equations: \( x - y = 2 \) and \( x + y = 22 \).
05

Solve the System of Equations

Add the equations \( x - y = 2 \) and \( x + y = 22 \): \[ (x - y) + (x + y) = 2 + 22 \implies 2x = 24 \implies x = 12. \] Now substitute \(x = 12\) back into \(x + y = 22\): \[ 12 + y = 22 \implies y = 10. \]

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Key Concepts

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

Difference of Squares
In algebra, the difference of squares is a special formula that can simplify expressions and solve equations. The formula is: \[ x^2 - y^2 = (x + y)(x - y) \] This means that if you have the square of two numbers subtracted from each other, you can represent it as the product of the sum and the difference of those two numbers. Let's consider our example, where you have the equation: \[ x^2 - y^2 = 44 \] We know from the problem statement that \( x - y = 2 \). Substituting this into the difference of squares formula gives us: \[ 44 = (x + y)(2) \] We can then solve for \( x + y \). This approach simplifies solving equations where squares of the numbers are involved.
Systems of Equations
A system of equations is a set of two or more equations with the same variables. In our example, we created a system of equations from the given information: \[ x - y = 2 \] \[ x + y = 22 \] The goal is to find values for \( x \) and \( y \) that satisfy both equations. One common method to solve these is by addition or subtraction. In this problem, we added the two equations to eliminate variable \( y \): \[ (x - y) + (x + y) = 2 + 22 \ 2x = 24 \ x = 12 \] With \( x \) found, we substitute it back into one of the original equations to find \( y \). This step-by-step process is very effective for efficiently solving systems of linear equations.
Linear Equations
Linear equations are equations of the first order, meaning they involve only the first power of the variable (e.g., \( x \) or \( y \)). These equations graph as straight lines. In our example, the linear equations were: \[ x - y = 2 \] \[ x + y = 22 \] Each equation represents a line on a graph. The solution to the system of equations is the point where these two lines intersect. Solving the equations simultaneously gives us this intersection point. For instance, we found: \[ x = 12 \] \[ y = 10 \] These values represent the intersection of the two lines described by our original linear equations, meaning 12 and 10 are the numbers that solve the initial algebra problem.

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Most popular questions from this chapter

Determine the number of solutions to the system of equations. $$ \begin{aligned} y &=2^{x+1} \\ -1+\log _{2} y &=x \end{aligned} $$

Use a graphing utility to approximate the solution(s) to the system of equations. Round the coordinates to 3 decimal places. $$ \begin{array}{l} y=-0.7 x+4 \\ y=\ln x \end{array} $$

Use a graphing utility to graph the solution set to the system of inequalities. \(y<\frac{4}{x^{2}+1}\) \(y>\frac{-2}{x^{2}+0.5}\)

A manufacturer produces two models of a gas grill. Grill A requires 1 hr for assembly and \(0.4 \mathrm{hr}\) for packaging. Grill \(B\) requires 1.2 hr for assembly and 0.6 hr for packaging. The production information and profit for each grill are given in the table. (See Example 4\()\) $$ \begin{array}{|l|c|c|c|} \hline & \text { Assembly } & \text { Packaging } & \text { Profit } \\ \hline \text { Grill A } & 1 \mathrm{hr} & 0.4 \mathrm{hr} & \$ 90 \\ \hline \text { Grill B } & 1.2 \mathrm{hr} & 0.6 \mathrm{hr} & \$ 120 \\ \hline \end{array} $$ The manufacturer has \(1200 \mathrm{hr}\) of labor available for assembly and \(540 \mathrm{hr}\) of labor available for packaging. a. Determine the number of grill A units and the number of grill B units that should be produced to maximize profit assuming that all grills will be sold. b. What is the maximum profit under these constraints? c. If the profit on grill A units is $$\$ 110$$ and the profit on grill \(\underline{B}\) units is unchanged, how many of each type of grill unit should the manufacturer produce to maximize profit?

Two particles begin at the same point and move at different speeds along a circular path of circumference \(280 \mathrm{ft}\). Moving in opposite directions, they pass in \(10 \mathrm{sec} .\) Moving in the same direction, they pass in \(70 \mathrm{sec} .\) Find the speed of each particle.
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