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Chapter 4: Problem 34

Write a system of equations modeling the given conditions. Then solve the system by the substitution method and find the two numbers. The sum of two numbers is \(62 .\) One number is 12 more than the other. Find the numbers.

Short Answer

Expert verified
The two numbers are 37 and 25.

Step by step solution

01

Setup the System of Equations

Let's call the two numbers \(x\) and \(y\). Given that the sum of the two numbers is 62, this can be represented as the equation \(x + y = 62 \). Given that one number is 12 more than the other, this can be represented as the equation \(x = y + 12\). Thus, the system of equations to solve is \(\{x + y = 62, x = y + 12\}\)
02

Substitute \(y + 12\) for \(x\) in the First Equation

Replace \(x\) in the first equation with the expression from the second equation (\(y + 12\)) to get the new equation: \(y + 12 + y = 62\).
03

Solve for \(y\)

Simplify and solve the equation for \(y\). \(2y + 12 = 62\) Subtract 12 from both sides to get \(2y = 50\). Then, divide by 2 to find that \(y = 25\).
04

Substitute \(y = 25\) Into the Second Equation

Substitute \(y = 25\) into the second equation \(x = y + 12\) to get \(x = 25 + 12\), hence \(x = 37\).
05

Check the Solution

Verify that these values work in the original equations. Substituting \(x = 37\) and \(y = 25\) into both initial equations shows that they indeed are solutions.

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Most popular questions from this chapter

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} x=5-3 y \\ 2 x+6 y=10 \end{array}\right.$$

Read Exercise \(72 .\) Then use a graphing utility to solve the systems. $$\left\\{\begin{array}{l}x+2 y=4 \\ x-y=4\end{array}\right.$$

In Lewis Carroll's Through the Looking Glass, the Following dialogue takes place: Tweedledum (to Tweedledee): The sum of your weight and twice mine is 361 pounds. Tweedledee (to Tweedledum): Contrawise, the sum of your weight and twice mine is 362 pounds. Find the weight of each of the two characters. (IMAGE CAN NOT COPY)

In Exercises \(80-83,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equations \(y=x-1\) and \(x=y+1\) are dependent.

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 2(x+2 y) &=6 \\ 3(x+2 y-3) &=0 \end{aligned}\right.$$
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