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Magazine Chapter 9: Problem 49
Solve each problem using a system of equations in two variables. Unknown Numbers Find two numbers whose sum is 17 and whose product is 42 .
Short Answer
Expert verified
The two numbers are 14 and 3.
Step by step solution
01
Define the Variables
Let the two unknown numbers be represented by the variables \( x \) and \( y \).
02
Create the System of Equations
Based on the problem, the sum of the numbers is 17 and their product is 42. Therefore, we have two equations:\( x + y = 17 \) and \( xy = 42 \).
03
Solve the First Equation for One Variable
Solve the equation \( x + y = 17 \) for one of the variables, for example, \( y \). Thus, \( y = 17 - x \).
04
Substitute the Expression into the Second Equation
Substitute \( y = 17 - x \) into the second equation \( xy = 42 \). This gives us:\[ x(17 - x) = 42 \].
05
Solve the Quadratic Equation
Expand the equation to get a quadratic form: \[ 17x - x^2 = 42 \]. Rearrange it as follows: \[ x^2 - 17x + 42 = 0 \]. Solve the quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -17 \), and \( c = 42 \). This results in: \[ x = \frac{17 \pm \sqrt{17^2 - 4(1)(42)}}{2(1)} = \frac{17 \pm \sqrt{289 - 168}}{2} = \frac{17 \pm \sqrt{121}}{2} = \frac{17 \pm 11}{2} \].
06
Find the Solutions for \( x \)
This gives us two potential solutions for \( x \): \[ x = \frac{17 + 11}{2} = 14 \] and \[ x = \frac{17 - 11}{2} = 3 \].
07
Find the Corresponding Values for \( y \)
Using \( y = 17 - x \), for both values of \( x \): If \( x = 14 \), then \( y = 17 - 14 = 3 \).If \( x = 3 \), then \( y = 17 - 3 = 14 \).
08
State the Solution
The two numbers are 14 and 3. Thus, the solutions are the pairs (14, 3) and (3, 14).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation
A quadratic equation is a second-order polynomial equation in a single variable, typically represented as \(ax^2 + bx + c = 0\). The solutions to a quadratic equation are given by the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In the context of our exercise, we derived a quadratic equation from the system of equations and used this formula to find the values of \(x\). It's important to understand that the quadratic formula helps us solve the equation by finding the points where the quadratic polynomial intersects the x-axis. This involves:
- Identifying the coefficients \(a\), \(b\), and \(c\).
- Calculating the discriminant \(b^2 - 4ac\).
- Substituting these values into the quadratic formula to find the solutions.
- Identifying the coefficients \(a\), \(b\), and \(c\).
- Calculating the discriminant \(b^2 - 4ac\).
- Substituting these values into the quadratic formula to find the solutions.
Solving Systems
A system of equations consists of two or more equations with the same set of variables. The goal when solving systems of equations is to find the set of values for the variables that satisfies all equations simultaneously. In our exercise, we have a system:
- \(x + y = 17\)
- \(xy = 42\)
To solve this system, we used the substitution method. We solved the first equation for one variable (\(y = 17 - x\)) and then substituted this expression into the second equation. This allowed us to reduce the problem to a single-variable quadratic equation, which we then solved using the quadratic formula. This step-wise approach simplifies complex problems and makes them more manageable.
- \(x + y = 17\)
- \(xy = 42\)
To solve this system, we used the substitution method. We solved the first equation for one variable (\(y = 17 - x\)) and then substituted this expression into the second equation. This allowed us to reduce the problem to a single-variable quadratic equation, which we then solved using the quadratic formula. This step-wise approach simplifies complex problems and makes them more manageable.
Two Variables
Dealing with two variables often involves understanding their relationships through equations. In our exercise, the two unknown numbers are our variables. The given conditions (their sum is 17 and their product is 42) lead to a system of linear and quadratic equations. By representing these numbers as \(x\) and \(y\), we can:
- Use algebraic techniques like substitution or elimination to simplify the system.
- Introduce one equation into another to either directly solve for one variable or reduce the system to a simpler form.
Understanding the interplay between multiple variables in this manner provides a structured way to tackle more complex mathematical problems involving higher dimensions and relationships.
- Use algebraic techniques like substitution or elimination to simplify the system.
- Introduce one equation into another to either directly solve for one variable or reduce the system to a simpler form.
Understanding the interplay between multiple variables in this manner provides a structured way to tackle more complex mathematical problems involving higher dimensions and relationships.
