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Chapter 4: Problem 33

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 \(81 .\) One number is 41 more than the other. Find the numbers.

Short Answer

Expert verified
The two numbers are 61 and 20.

Step by step solution

01

Convert Word Problems into Equations

Let the two numbers be \(x\) and \(y\). From the problem, we know that \(x + y = 81\) (because their sum is 81) and \(x = y + 41\) (because one number is 41 more than the other).
02

Substitute the Value from the Second Equation into the First

Substituting \(x\) from the second equation into the first equation, we get \(y + 41 + y = 81\).
03

Solve for y

By simplifying the equation, consolidating like terms, we get \(2y + 41 = 81\). Hence, by subtracting 41 from both sides, we get \(2y = 81 - 41 = 40\). Dividing by 2 we get \(y = 40 / 2 = 20\).
04

Substitute y into the Second Original Equation

By substituting \(y = 20\) into the equation \(x = y + 41\), we get \(x = 20 + 41 = 61\).
05

Verify the Equation

We will verify if \(x = 61\) and \(y = 20\) satisfy the first original equation \(x + y = 81\). By inserting values we find the equation \(81 = 81\) to be true thus verifying the calculated solution.

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Most popular questions from this chapter

Perform the indicated operation. \(-3+(-9)\) (Section \(1.7,\) Table 1.7 )

Read Exercise \(72 .\) Then use a graphing utility to solve the systems. $$\left\\{\begin{array}{l}y=2 x+2 \\ y=-2 x+6\end{array}\right.$$

Verify your solutions to any five exercises from Exercises 11 through 36 by using a graphing utility to graph the two equations in the system in the same viewing rectangle. After entering the two equations, one as \(y_{1}\) and the other as \(y_{2},\) and graphing them, use the \([\text { INTERSECTION }]\) feature to find the system's solution. (It may first be necessary to solve the equations for \(y\) before entering them.)

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When using the addition method, how can you tell if a system of linear equations has no solution?
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