/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Set up and solve an appropriate ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 37

Set up and solve an appropriate system of linear equations to answer the questions. Grace is three times as old as Hans, but in 5 years she will be twice as old as Hans is then. How old are they now?

Short Answer

Expert verified
Hans is 5 years old, and Grace is 15 years old.

Step by step solution

01

Define Variables

Let's define the variables. Let \( G \) represent Grace's current age and \( H \) represent Hans's current age.
02

Set Up Equations

From the problem, we have two pieces of information:1. Grace is three times as old as Hans: \( G = 3H \).2. In 5 years, Grace will be twice as old as Hans: \( G + 5 = 2(H + 5) \).
03

Simplify the Second Equation

Simplify the second equation:\[ G + 5 = 2(H + 5) \]\[ G + 5 = 2H + 10 \]\[ G = 2H + 5 \].
04

Solve the System of Equations

We now solve the system given by the equations:1. \( G = 3H \)2. \( G = 2H + 5 \)Substitute the expression for \( G \) from the first equation into the second:\[ 3H = 2H + 5 \].Now, solve for \( H \):\[ 3H - 2H = 5 \]\[ H = 5 \].
05

Calculate Grace's Age

Use the value of \( H \) to find \( G \):\[ G = 3H = 3 \times 5 = 15 \].
06

Verify the Solution

Check that our solution satisfies both original statements:1. Grace is three times as old as Hans: \( 15 = 3 \times 5 \).2. In 5 years, Grace will be twice as old as Hans: \( 15 + 5 = 2 \times (5 + 5) \), which simplifies to \( 20 = 2 \times 10 = 20 \).Thus, both statements are satisfied with Hans being 5 and Grace being 15.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Age Problems
Age problems are a unique type of algebraic puzzle that involves people of various ages usually at different points in time. The key to solving these problems is to carefully translate the word problem into mathematical equations using variables. Commonly, age problems provide a current age or imply a relationship between ages, which can later be translated into equations.
For example, you may be told a person's age relative to someone else's, or how their age will compare at a future date. In our problem, we know two facts about Grace and Hans:
  • Grace's current age is three times that of Hans.
  • Five years later, she will only be twice as old as Hans.
Understanding and identifying these relationships are crucial for breaking the problem down into solvable steps using algebraic methods.
Solving Equations
Solving equations is the main process we use to find unknown values represented by variables. In age problems, solving a system of linear equations is the typical approach.
A system of linear equations is a collection of two or more equations with the same set of unknowns. Here, we set up two equations:
  • Grace is currently 3 times Hans's age, represented as: \( G = 3H \).
  • 5 years later, Grace will be twice as old as Hans: \( G + 5 = 2(H + 5) \).
Solving this system involves manipulation of these equations to isolate and find the values of the variables. First, simplify the second equation: \( G + 5 = 2H + 10 \) simplifies to \( G = 2H + 5 \).
Next, you solve the equations using your chosen algebraic method, ensuring you arrive at values for the variables that satisfy both equations and thus correctly solve the age problem.
Substitution Method
The substitution method is one of the most common techniques for solving a system of equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation.
In our exercise, we solve the equation \( G = 3H \) for \( G \), which directly provides expression for Grace's age. We then substitute \( G \) into the second equation \( G = 2H + 5 \), allowing us to solve for \( H \).
Here's how it works:
  • The first equation gives us \( G = 3H \).
  • Substituting into the second equation, we get \( 3H = 2H + 5 \).
  • Solving \( 3H = 2H + 5 \) gives \( H = 5 \).
Once we have the value for \( H \), we substitute back to find \( G = 15 \). This method is effective because it reduces the number of equations, making the system easier to handle.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Balance the chemical equation for each reaction. \(\mathrm{C}_{2} \mathrm{H}_{2} \mathrm{Cl}_{4}+\mathrm{Ca}(\mathrm{OH})_{2} \longrightarrow \mathrm{C}_{2} \mathrm{HCl}_{3}+\mathrm{CaCl}_{2}+\mathrm{H}_{2} \mathrm{O}\)

Use elementary row operations to reduce the given matrix to ( a) row echelon form and ( \(b\) ) reduced row echelon form. \(\left[\begin{array}{rrrr}2 & -4 & -2 & 6 \\ 3 & 1 & 6 & 6\end{array}\right]\)

Solve the given system of equations using either Gaussian or Gauss-Jordan elimination. \(2 r+s=3\) \(4 r+s=7\) \(2 r+5 s=-1\)

Set up and solve an appropriate system of linear equations to answer the questions. Following are two useful formulas for the sums of powers of consecutive natural numbers: $$1+2+\cdots+n=\frac{n(n+1)}{2}$$ and $$1^{2}+2^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$$ The validity of these formulas for all values of \(n \geq 1\) (or even \(n \geq 0\) ) can be established using mathematical induction (see Appendix B). One way to make an educated guess as to what the formulas are, though, is to observe that we can rewrite the two formulas above as $$\frac{1}{2} n^{2}+\frac{1}{2} \quad \text { and } \quad \frac{1}{3} n^{3}+\frac{1}{2} n^{2}+\frac{1}{6} n$$ respectively. This leads to the conjecture that the sum of pth powers of the first \(n\) natural numbers is a polynomial of degree \(p+1\) in the variable \(n\). Assume that \(1^{2}+2^{2}+\cdots+n^{2}=a n^{3}+b n^{2}+c n+d\) Find \(a, b, c,\) and \(d .[\text { Hint: It is legitimate to use } n=0\) What is the left-hand side in that case?]

Suppose the coal and steel industries form an open economy. Every \(\$ 1\) produced by the coal industry requires \(\$ 0.15\) of coal and \(\$ 0.20\) of steel. Every \(\$ 1\) produced by steel requires \(\$ 0.25\) of coal and \(\$ 0.10\) of steel. Suppose that there is an annual outside demand for \(\$ 45\) million of coal and \(\$ 124\) million of steel. (a) How much should each industry produce to satisfy the demands? (b) If the demand for coal decreases by \(\$ 5\) million per year while the demand for steel increases by \$6 million per year, how should the coal and steel industries adjust their production?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.