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Chapter 4: Problem 17

Solve each system of equations by the substitution method. See Examples I through 6 $$ \left\\{\begin{array}{l} {4 x+2 y=5} \\ {-2 x=y+4} \end{array}\right. $$

Short Answer

Expert verified
The system has no solution as it is inconsistent.

Step by step solution

01

Solve one equation for one variable

The second equation is simpler, so we'll solve it for \( y \).\[-2x = y + 4\]\[y = -2x - 4\]
02

Substitute the expression into the other equation

Substitute \( y = -2x - 4 \) into the first equation\[4x + 2(-2x - 4) = 5\]
03

Simplify and solve for \( x \)

Distribute the 2 into the parenthesis:\[4x - 4x - 8 = 5\]\[-8 = 5\]. This can't be true, indicating no solution with this substitution.
04

Verify with a logical check

Since \(-8 = 5\) is not logically possible, the system of equations is inconsistent. Therefore, there is no solution.

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Key Concepts

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

Understanding Systems of Equations
In algebra, a system of equations consists of two or more equations with multiple variables. The goal is to find a set of values for the variables that satisfies all equations simultaneously.
A simple system might involve two equations with two variables, such as our example:
  • \( 4x + 2y = 5 \)
  • \( -2x = y + 4 \)
A solution to this system is a pair \(x, y\) that makes both equations true at the same time. There are often multiple methods to solve these systems, such as graphing, substitution, or elimination. In this exercise, we use the substitution method, which is particularly useful for finding solutions by isolating one variable.
Identifying an Inconsistent System
An inconsistent system of equations is one that has no solution. This occurs when the equations represent parallel lines that never intersect, thus never sharing any common points.
During the solution process, if you reach a false statement, such as \( -8 = 5 \), it indicates that the system is inconsistent. For our example, upon substituting and simplifying the equations, we encountered this condition, revealing that no values of \( x \) and \( y \) can satisfy both equations simultaneously.
Recognizing an inconsistent system is crucial because it saves time and redirects your approach to analyzing the relationship between the equations.
Applying Algebraic Solutions to Systems
To solve equations algebraically, we manipulate them using operations that maintain their equality. For solving systems, like ours, we often start by expressing one variable in terms of another using substitution. In our example, we express \( y \) in terms of \( x \) as follows:
  • From the second equation: \( y = -2x - 4 \)
This expression substitutes back into the first equation, allowing us to solve for one variable, quite efficiently.
Algebraic solutions require carefully maintaining balance across the equation, ensuring every step is logically derived from the previous one.
Steps in Solving Equations
The process of solving equations involves a sequence of logical steps, especially when employing the substitution method (our method of choice in this exercise). Below is a brief breakdown of the steps:
  • Step 1: Solve one equation for one of the variables. For example, find \( y \) in terms of \( x \).
  • Step 2: Substitute the expression from step 1 into the other equation. This substitution allows you to solve for the remaining variable.
  • Step 3: Simplify and solve the resulting equation. If it results in a false statement, indicate that the system is inconsistent as in our exercise.
These steps offer a structured approach to tackling systems of equations. Not every system will have a solution, and identifying inconsistencies is as vital as finding an exact solution when possible.

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Most popular questions from this chapter

Elise Everly is preparing 15 liters of a \(25 \%\) saline solution. Elise has two other saline solutions with strengths of \(40 \%\) and \(10 \% .\) Find the amount of \(40 \%\) solution and the amount of \(10 \%\) solution she should mix to get 15 liters of a \(25 \%\) solution. $$ \begin{array}{|c|c|c|} \hline \text { Concentration } & {\text {Liters of }} & {\text {Liters of }} \\\ {\text { Rate }} & {\text { Solution }} & {\text { Pure Salt }} \\ \hline 0.40 & {x} & {0.40 x} \\ \hline 0.10 & {y} & {?} \\ \hline 0.25 & {15} & {?} \\ \hline \end{array} $$

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Solve each system of linear equations by graphing. See Examples 3 through 6 $$ \left\\{\begin{array}{l} {y=x-2} \\ {y=2 x+3} \end{array}\right. $$

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Pratap Puri rowed 18 miles down the Delaware River in 2 hours, but the return trip took him \(4 \frac{1}{2}\) hours. Find the rate Pratap can row in still water, and find the rate of the current. Let \(x=\) rate Pratap can row in still water and \(y=\) rate of the current. (TABLE CANNOT COPY)
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