/*! 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 44 Solve by substitution. $$ \b... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 44

Solve by substitution. $$ \begin{aligned} &c=2 b-1 \\ &b=2 c+1 \end{aligned} $$

Short Answer

Expert verified
The solutions are \( b = \frac{1}{3} \) and \( c = -\frac{1}{3} \).

Step by step solution

01

Solve one equation for a variable

Start by solving the first equation, \( c = 2b - 1 \), for one variable. In this case, we already have \( c \) expressed in terms of \( b \).
02

Substitute the expression into the second equation

Substitute the expression for \( c \) from the first equation into the second equation. Starting with the second equation: \( b = 2c + 1 \), substitute \( c = 2b - 1 \) to get: \( b = 2(2b - 1) + 1 \).
03

Simplify the equation

Now simplify the equation: \( b = 2(2b - 1) + 1 \) Expand the terms: \( b = 4b - 2 + 1 \) Simplify further: \( b = 4b - 1 \)
04

Solve for the variable

Rearrange the equation to solve for \( b \): \( b - 4b = -1 \) Combine like terms: \( -3b = -1 \) Divide both sides by -3: \( b = \frac{1}{3} \)
05

Substitute back to find the other variable

Use the value of \( b \) to find \( c \). Substitute \( b = \frac{1}{3} \) back into the equation \( c = 2b - 1 \): \( c = 2\left(\frac{1}{3}\right) - 1 \) Simplify \( c = \frac{2}{3} - 1 \) Convert to a common denominator: \( c = \frac{2}{3} - \frac{3}{3} \) Subtract: \( c = -\frac{1}{3} \)

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Key Concepts

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

Understanding the Substitution Method
The substitution method is a powerful tool for solving systems of linear equations. It involves expressing one variable in terms of another and then substituting this expression into another equation. This method helps us reduce the system to a single equation with one variable. For example, in the problem given, the first step expresses the variable 'c' in terms of 'b':
  • Equation: c = 2b - 1
This expression is then substituted into the second equation to solve for 'b'. Once 'b' is found, we substitute back to find 'c'. This step-by-step approach ensures we solve the system accurately.
Linear Equations Explained
Linear equations are equations of the first degree, meaning their highest exponent is one. They represent straight lines when graphed on a coordinate plane. In the given problem, we have two linear equations:
  • c = 2b - 1
  • b = 2c + 1
These equations can be manipulated using algebraic methods to find the values of 'b' and 'c'. Understanding the nature of linear equations is crucial because it allows us to predict and interpret their solutions. They often form the basis of more complex systems, making them fundamental to algebra.
Variable Elimination for Simplification
Variable elimination is a core concept in solving systems of linear equations. It helps us isolate one variable at a time, making the system simpler to solve. In the problem, once we substitute 'c' from the first equation into the second, we get a new equation with only 'b':
  • Starting equation: b = 2(2b - 1) + 1
  • After expanding: b = 4b - 2 + 1
This process eliminates 'c' from the second equation, allowing us to solve for 'b'. Once 'b' is determined, it is then used to find 'c'. This method reduces the complexity of solving two equations with two unknowns.

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Most popular questions from this chapter

A karat describes the percent gold in an alloy (a mixture of metals). $$ \begin{array}{|c|c|} \hline \text { Name of alloy } & \text { Percent gold } \\ \hline \text { 10-karat gold } & 41.7 \% \\ \text { 14-karat gold } & 58.3 \% \\ \text { 18-karat gold } & 75 \% \\ \text { 20-karat gold } & 83.3 \% \\ \text { 24-karat gold } & 100 \% \\ \hline \end{array} $$ If 9 oz of 14-karat gold jewelry is melted down, find the amount of 20 -karat gold to add to the melted jewelry to create a new alloy that is \(18-k\) arat gold. Find the amount of the new alloy. Round to the nearest tenth.

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The perimeter of a rectangle is \(78 \mathrm{in}\). The length is \(9 \mathrm{in}\). longer than the width. Find the length and the width.

A fisheries department needs a total of 5760 kokanee trout and rainbow trout for stocking lakes. They want three times as many kokanee trout as rainbow trout. Find the number of kokanee trout and the number of rainbow trout that they need.

A truck leaves a town traveling at a constant speed of \(\frac{55 \mathrm{mi}}{1 \mathrm{hr}}\). After \(20 \mathrm{~min}\), a car follows the same route traveling at a constant speed of \(\frac{60 \mathrm{mi}}{1 \mathrm{hr}}\). Find the time in minutes when the car will catch up with the truck. Find the distance traveled by the truck and the distance traveled by the car.
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